Finite Sequence Windows, Extensions, and Ordered Differences
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This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.
batch-20261011-advance-20261015-r05-sequence-foundations · 2026-10-11 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
1. Finite Sequence Query Bounds before Evaluation
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. L=0; only 0≤n<L is allowed for a_n=1+(2)n. Ordered candidate indices=[-1, 0, 1]. Give (L, index-eligibility mask, legal term values or invalid-index).
View answer and explanation
(0, [No, No, No], [invalid-index, invalid-index, invalid-index])
Check the finite index bound before substitution; an outside index does not acquire a term merely because the affine expression could be evaluated there. The exact result, retaining the requested order, is (0, [No, No, No], [invalid-index, invalid-index, invalid-index]).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-window-boundary-queries · candidate
2. Finite Sequence Query Bounds before Evaluation
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. L=1; only 0≤n<L is allowed for a_n=0+(0)n. Ordered candidate indices=[0, 1, -1]. Give (L, index-eligibility mask, legal term values or invalid-index).
View answer and explanation
(1, [Yes, No, No], [0, invalid-index, invalid-index])
Check the finite index bound before substitution; an outside index does not acquire a term merely because the affine expression could be evaluated there. The exact result, retaining the requested order, is (1, [Yes, No, No], [0, invalid-index, invalid-index]).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-window-boundary-queries · candidate
3. Finite Sequence Query Bounds before Evaluation
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. L=4; only 0≤n<L is allowed for a_n=-10+(1)n. Ordered candidate indices=[0, 3, 4]. Give (L, index-eligibility mask, legal term values or invalid-index).
View answer and explanation
(4, [Yes, Yes, No], [-10, -7, invalid-index])
Check the finite index bound before substitution; an outside index does not acquire a term merely because the affine expression could be evaluated there. The exact result, retaining the requested order, is (4, [Yes, Yes, No], [-10, -7, invalid-index]).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-window-boundary-queries · candidate
4. Finite Sequence Query Bounds before Evaluation
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. L=4; only 0≤n<L is allowed for a_n=-9+(-1)n. Ordered candidate indices=[0, 3, 4]. Give (L, index-eligibility mask, legal term values or invalid-index).
View answer and explanation
(4, [Yes, Yes, No], [-9, -12, invalid-index])
Check the finite index bound before substitution; an outside index does not acquire a term merely because the affine expression could be evaluated there. The exact result, retaining the requested order, is (4, [Yes, Yes, No], [-9, -12, invalid-index]).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-window-boundary-queries · candidate
5. Finite Sequence Query Bounds before Evaluation
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. L=4; only 0≤n<L is allowed for a_n=-8+(0)n. Ordered candidate indices=[0, 3, 4]. Give (L, index-eligibility mask, legal term values or invalid-index).
View answer and explanation
(4, [Yes, Yes, No], [-8, -8, invalid-index])
Check the finite index bound before substitution; an outside index does not acquire a term merely because the affine expression could be evaluated there. The exact result, retaining the requested order, is (4, [Yes, Yes, No], [-8, -8, invalid-index]).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-window-boundary-queries · candidate
6. Finite Sequence Query Bounds before Evaluation
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. L=4; only 0≤n<L is allowed for a_n=-7+(1)n. Ordered candidate indices=[0, 3, 4]. Give (L, index-eligibility mask, legal term values or invalid-index).
View answer and explanation
(4, [Yes, Yes, No], [-7, -4, invalid-index])
Check the finite index bound before substitution; an outside index does not acquire a term merely because the affine expression could be evaluated there. The exact result, retaining the requested order, is (4, [Yes, Yes, No], [-7, -4, invalid-index]).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-window-boundary-queries · candidate
7. Finite Sequence Query Bounds before Evaluation
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. L=4; only 0≤n<L is allowed for a_n=-6+(-1)n. Ordered candidate indices=[0, 3, 4]. Give (L, index-eligibility mask, legal term values or invalid-index).
View answer and explanation
(4, [Yes, Yes, No], [-6, -9, invalid-index])
Check the finite index bound before substitution; an outside index does not acquire a term merely because the affine expression could be evaluated there. The exact result, retaining the requested order, is (4, [Yes, Yes, No], [-6, -9, invalid-index]).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-window-boundary-queries · candidate
8. Finite Sequence Query Bounds before Evaluation
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. L=4; only 0≤n<L is allowed for a_n=-5+(0)n. Ordered candidate indices=[0, 3, 4]. Give (L, index-eligibility mask, legal term values or invalid-index).
View answer and explanation
(4, [Yes, Yes, No], [-5, -5, invalid-index])
Check the finite index bound before substitution; an outside index does not acquire a term merely because the affine expression could be evaluated there. The exact result, retaining the requested order, is (4, [Yes, Yes, No], [-5, -5, invalid-index]).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-window-boundary-queries · candidate
9. Finite Sequence Query Bounds before Evaluation
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. L=4; only 0≤n<L is allowed for a_n=-4+(1)n. Ordered candidate indices=[0, 3, 4]. Give (L, index-eligibility mask, legal term values or invalid-index).
View answer and explanation
(4, [Yes, Yes, No], [-4, -1, invalid-index])
Check the finite index bound before substitution; an outside index does not acquire a term merely because the affine expression could be evaluated there. The exact result, retaining the requested order, is (4, [Yes, Yes, No], [-4, -1, invalid-index]).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-window-boundary-queries · candidate
10. Finite Sequence Query Bounds before Evaluation
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. L=4; only 0≤n<L is allowed for a_n=-3+(-1)n. Ordered candidate indices=[0, 3, 4]. Give (L, index-eligibility mask, legal term values or invalid-index).
View answer and explanation
(4, [Yes, Yes, No], [-3, -6, invalid-index])
Check the finite index bound before substitution; an outside index does not acquire a term merely because the affine expression could be evaluated there. The exact result, retaining the requested order, is (4, [Yes, Yes, No], [-3, -6, invalid-index]).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-window-boundary-queries · candidate
11. Finite Sequence Query Bounds before Evaluation
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. L=4; only 0≤n<L is allowed for a_n=-2+(0)n. Ordered candidate indices=[0, 3, 4]. Give (L, index-eligibility mask, legal term values or invalid-index).
View answer and explanation
(4, [Yes, Yes, No], [-2, -2, invalid-index])
Check the finite index bound before substitution; an outside index does not acquire a term merely because the affine expression could be evaluated there. The exact result, retaining the requested order, is (4, [Yes, Yes, No], [-2, -2, invalid-index]).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-window-boundary-queries · candidate
12. Finite Sequence Query Bounds before Evaluation
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. L=4; only 0≤n<L is allowed for a_n=-1+(1)n. Ordered candidate indices=[0, 3, 4]. Give (L, index-eligibility mask, legal term values or invalid-index).
View answer and explanation
(4, [Yes, Yes, No], [-1, 2, invalid-index])
Check the finite index bound before substitution; an outside index does not acquire a term merely because the affine expression could be evaluated there. The exact result, retaining the requested order, is (4, [Yes, Yes, No], [-1, 2, invalid-index]).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-window-boundary-queries · candidate
13. Finite Sequence Query Bounds before Evaluation
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. L=4; only 0≤n<L is allowed for a_n=0+(-1)n. Ordered candidate indices=[0, 3, 4]. Give (L, index-eligibility mask, legal term values or invalid-index).
View answer and explanation
(4, [Yes, Yes, No], [0, -3, invalid-index])
Check the finite index bound before substitution; an outside index does not acquire a term merely because the affine expression could be evaluated there. The exact result, retaining the requested order, is (4, [Yes, Yes, No], [0, -3, invalid-index]).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-window-boundary-queries · candidate
14. Finite Sequence Query Bounds before Evaluation
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. L=4; only 0≤n<L is allowed for a_n=1+(0)n. Ordered candidate indices=[0, 3, 4]. Give (L, index-eligibility mask, legal term values or invalid-index).
View answer and explanation
(4, [Yes, Yes, No], [1, 1, invalid-index])
Check the finite index bound before substitution; an outside index does not acquire a term merely because the affine expression could be evaluated there. The exact result, retaining the requested order, is (4, [Yes, Yes, No], [1, 1, invalid-index]).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-window-boundary-queries · candidate
15. Finite Sequence Query Bounds before Evaluation
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. L=4; only 0≤n<L is allowed for a_n=2+(1)n. Ordered candidate indices=[0, 3, 4]. Give (L, index-eligibility mask, legal term values or invalid-index).
View answer and explanation
(4, [Yes, Yes, No], [2, 5, invalid-index])
Check the finite index bound before substitution; an outside index does not acquire a term merely because the affine expression could be evaluated there. The exact result, retaining the requested order, is (4, [Yes, Yes, No], [2, 5, invalid-index]).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-window-boundary-queries · candidate
16. Finite Sequence Query Bounds before Evaluation
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. L=4; only 0≤n<L is allowed for a_n=3+(-1)n. Ordered candidate indices=[0, 3, 4]. Give (L, index-eligibility mask, legal term values or invalid-index).
View answer and explanation
(4, [Yes, Yes, No], [3, 0, invalid-index])
Check the finite index bound before substitution; an outside index does not acquire a term merely because the affine expression could be evaluated there. The exact result, retaining the requested order, is (4, [Yes, Yes, No], [3, 0, invalid-index]).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-window-boundary-queries · candidate
17. Finite Sequence Query Bounds before Evaluation
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. L=4; only 0≤n<L is allowed for a_n=4+(0)n. Ordered candidate indices=[0, 3, 4]. Give (L, index-eligibility mask, legal term values or invalid-index).
View answer and explanation
(4, [Yes, Yes, No], [4, 4, invalid-index])
Check the finite index bound before substitution; an outside index does not acquire a term merely because the affine expression could be evaluated there. The exact result, retaining the requested order, is (4, [Yes, Yes, No], [4, 4, invalid-index]).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-window-boundary-queries · candidate
18. Finite Sequence Query Bounds before Evaluation
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. L=4; only 0≤n<L is allowed for a_n=5+(1)n. Ordered candidate indices=[0, 3, 4]. Give (L, index-eligibility mask, legal term values or invalid-index).
View answer and explanation
(4, [Yes, Yes, No], [5, 8, invalid-index])
Check the finite index bound before substitution; an outside index does not acquire a term merely because the affine expression could be evaluated there. The exact result, retaining the requested order, is (4, [Yes, Yes, No], [5, 8, invalid-index]).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-window-boundary-queries · candidate
19. Finite Sequence Query Bounds before Evaluation
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. L=4; only 0≤n<L is allowed for a_n=6+(-1)n. Ordered candidate indices=[0, 3, 4]. Give (L, index-eligibility mask, legal term values or invalid-index).
View answer and explanation
(4, [Yes, Yes, No], [6, 3, invalid-index])
Check the finite index bound before substitution; an outside index does not acquire a term merely because the affine expression could be evaluated there. The exact result, retaining the requested order, is (4, [Yes, Yes, No], [6, 3, invalid-index]).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-window-boundary-queries · candidate
20. Finite Sequence Query Bounds before Evaluation
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. L=4; only 0≤n<L is allowed for a_n=7+(0)n. Ordered candidate indices=[0, 3, 4]. Give (L, index-eligibility mask, legal term values or invalid-index).
View answer and explanation
(4, [Yes, Yes, No], [7, 7, invalid-index])
Check the finite index bound before substitution; an outside index does not acquire a term merely because the affine expression could be evaluated there. The exact result, retaining the requested order, is (4, [Yes, Yes, No], [7, 7, invalid-index]).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-window-boundary-queries · candidate
21. Finite Sequence Query Bounds before Evaluation
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. L=4; only 0≤n<L is allowed for a_n=8+(1)n. Ordered candidate indices=[0, 3, 4]. Give (L, index-eligibility mask, legal term values or invalid-index).
View answer and explanation
(4, [Yes, Yes, No], [8, 11, invalid-index])
Check the finite index bound before substitution; an outside index does not acquire a term merely because the affine expression could be evaluated there. The exact result, retaining the requested order, is (4, [Yes, Yes, No], [8, 11, invalid-index]).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-window-boundary-queries · candidate
22. Finite Sequence Query Bounds before Evaluation
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. L=4; only 0≤n<L is allowed for a_n=9+(-1)n. Ordered candidate indices=[0, 3, 4]. Give (L, index-eligibility mask, legal term values or invalid-index).
View answer and explanation
(4, [Yes, Yes, No], [9, 6, invalid-index])
Check the finite index bound before substitution; an outside index does not acquire a term merely because the affine expression could be evaluated there. The exact result, retaining the requested order, is (4, [Yes, Yes, No], [9, 6, invalid-index]).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-window-boundary-queries · candidate
23. Finite Sequence Query Bounds before Evaluation
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. L=4; only 0≤n<L is allowed for a_n=10+(0)n. Ordered candidate indices=[0, 3, 4]. Give (L, index-eligibility mask, legal term values or invalid-index).
View answer and explanation
(4, [Yes, Yes, No], [10, 10, invalid-index])
Check the finite index bound before substitution; an outside index does not acquire a term merely because the affine expression could be evaluated there. The exact result, retaining the requested order, is (4, [Yes, Yes, No], [10, 10, invalid-index]).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-window-boundary-queries · candidate
24. Finite Sequence Query Bounds before Evaluation
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. L=4; only 0≤n<L is allowed for a_n=11+(1)n. Ordered candidate indices=[0, 3, 4]. Give (L, index-eligibility mask, legal term values or invalid-index).
View answer and explanation
(4, [Yes, Yes, No], [11, 14, invalid-index])
Check the finite index bound before substitution; an outside index does not acquire a term merely because the affine expression could be evaluated there. The exact result, retaining the requested order, is (4, [Yes, Yes, No], [11, 14, invalid-index]).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-window-boundary-queries · candidate
25. Finite Sequence Query Bounds before Evaluation
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. L=4; only 0≤n<L is allowed for a_n=12+(-1)n. Ordered candidate indices=[0, 3, 4]. Give (L, index-eligibility mask, legal term values or invalid-index).
View answer and explanation
(4, [Yes, Yes, No], [12, 9, invalid-index])
Check the finite index bound before substitution; an outside index does not acquire a term merely because the affine expression could be evaluated there. The exact result, retaining the requested order, is (4, [Yes, Yes, No], [12, 9, invalid-index]).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-window-boundary-queries · candidate
26. Ordered Recurrence Window and Adjacent Differences
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. a_0=0, a_(n+1)=a_n+(1), start=0, length=0. Give (window indices, term values, adjacent differences); length=0 gives three empty lists.
View answer and explanation
([], [], [])
Apply the declared recurrence from its given initial term and record the ordered window; the index list and repetitions remain meaningful. The exact result, retaining the requested order, is ([], [], []).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-affine-recurrence-window · candidate
27. Ordered Recurrence Window and Adjacent Differences
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. a_0=0, a_(n+1)=a_n+(7), start=0, length=1. Give (window indices, term values, adjacent differences); length=0 gives three empty lists.
View answer and explanation
([0], [0], [])
Apply the declared recurrence from its given initial term and record the ordered window; the index list and repetitions remain meaningful. The exact result, retaining the requested order, is ([0], [0], []).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-affine-recurrence-window · candidate
28. Ordered Recurrence Window and Adjacent Differences
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. a_0=-10, a_(n+1)=a_n+(1), start=0, length=3. Give (window indices, term values, adjacent differences); length=0 gives three empty lists.
View answer and explanation
([0, 1, 2], [-10, -9, -8], [1, 1])
Apply the declared recurrence from its given initial term and record the ordered window; the index list and repetitions remain meaningful. The exact result, retaining the requested order, is ([0, 1, 2], [-10, -9, -8], [1, 1]).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-affine-recurrence-window · candidate
29. Ordered Recurrence Window and Adjacent Differences
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. a_0=-9, a_(n+1)=a_n+(-1), start=1, length=3. Give (window indices, term values, adjacent differences); length=0 gives three empty lists.
View answer and explanation
([1, 2, 3], [-10, -11, -12], [-1, -1])
Apply the declared recurrence from its given initial term and record the ordered window; the index list and repetitions remain meaningful. The exact result, retaining the requested order, is ([1, 2, 3], [-10, -11, -12], [-1, -1]).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-affine-recurrence-window · candidate
30. Ordered Recurrence Window and Adjacent Differences
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. a_0=-8, a_(n+1)=a_n+(0), start=2, length=3. Give (window indices, term values, adjacent differences); length=0 gives three empty lists.
View answer and explanation
([2, 3, 4], [-8, -8, -8], [0, 0])
Apply the declared recurrence from its given initial term and record the ordered window; the index list and repetitions remain meaningful. The exact result, retaining the requested order, is ([2, 3, 4], [-8, -8, -8], [0, 0]).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-affine-recurrence-window · candidate
31. Ordered Recurrence Window and Adjacent Differences
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. a_0=-7, a_(n+1)=a_n+(1), start=3, length=3. Give (window indices, term values, adjacent differences); length=0 gives three empty lists.
View answer and explanation
([3, 4, 5], [-4, -3, -2], [1, 1])
Apply the declared recurrence from its given initial term and record the ordered window; the index list and repetitions remain meaningful. The exact result, retaining the requested order, is ([3, 4, 5], [-4, -3, -2], [1, 1]).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-affine-recurrence-window · candidate
32. Ordered Recurrence Window and Adjacent Differences
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. a_0=-6, a_(n+1)=a_n+(-1), start=4, length=3. Give (window indices, term values, adjacent differences); length=0 gives three empty lists.
View answer and explanation
([4, 5, 6], [-10, -11, -12], [-1, -1])
Apply the declared recurrence from its given initial term and record the ordered window; the index list and repetitions remain meaningful. The exact result, retaining the requested order, is ([4, 5, 6], [-10, -11, -12], [-1, -1]).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-affine-recurrence-window · candidate
33. Ordered Recurrence Window and Adjacent Differences
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. a_0=-5, a_(n+1)=a_n+(0), start=5, length=3. Give (window indices, term values, adjacent differences); length=0 gives three empty lists.
View answer and explanation
([5, 6, 7], [-5, -5, -5], [0, 0])
Apply the declared recurrence from its given initial term and record the ordered window; the index list and repetitions remain meaningful. The exact result, retaining the requested order, is ([5, 6, 7], [-5, -5, -5], [0, 0]).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-affine-recurrence-window · candidate
34. Ordered Recurrence Window and Adjacent Differences
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. a_0=-4, a_(n+1)=a_n+(1), start=6, length=3. Give (window indices, term values, adjacent differences); length=0 gives three empty lists.
View answer and explanation
([6, 7, 8], [2, 3, 4], [1, 1])
Apply the declared recurrence from its given initial term and record the ordered window; the index list and repetitions remain meaningful. The exact result, retaining the requested order, is ([6, 7, 8], [2, 3, 4], [1, 1]).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-affine-recurrence-window · candidate
35. Ordered Recurrence Window and Adjacent Differences
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. a_0=-3, a_(n+1)=a_n+(-1), start=7, length=3. Give (window indices, term values, adjacent differences); length=0 gives three empty lists.
View answer and explanation
([7, 8, 9], [-10, -11, -12], [-1, -1])
Apply the declared recurrence from its given initial term and record the ordered window; the index list and repetitions remain meaningful. The exact result, retaining the requested order, is ([7, 8, 9], [-10, -11, -12], [-1, -1]).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-affine-recurrence-window · candidate
36. Ordered Recurrence Window and Adjacent Differences
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. a_0=-2, a_(n+1)=a_n+(0), start=8, length=3. Give (window indices, term values, adjacent differences); length=0 gives three empty lists.
View answer and explanation
([8, 9, 10], [-2, -2, -2], [0, 0])
Apply the declared recurrence from its given initial term and record the ordered window; the index list and repetitions remain meaningful. The exact result, retaining the requested order, is ([8, 9, 10], [-2, -2, -2], [0, 0]).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-affine-recurrence-window · candidate
37. Ordered Recurrence Window and Adjacent Differences
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. a_0=-1, a_(n+1)=a_n+(1), start=9, length=3. Give (window indices, term values, adjacent differences); length=0 gives three empty lists.
View answer and explanation
([9, 10, 11], [8, 9, 10], [1, 1])
Apply the declared recurrence from its given initial term and record the ordered window; the index list and repetitions remain meaningful. The exact result, retaining the requested order, is ([9, 10, 11], [8, 9, 10], [1, 1]).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-affine-recurrence-window · candidate
38. Ordered Recurrence Window and Adjacent Differences
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. a_0=0, a_(n+1)=a_n+(-1), start=10, length=3. Give (window indices, term values, adjacent differences); length=0 gives three empty lists.
View answer and explanation
([10, 11, 12], [-10, -11, -12], [-1, -1])
Apply the declared recurrence from its given initial term and record the ordered window; the index list and repetitions remain meaningful. The exact result, retaining the requested order, is ([10, 11, 12], [-10, -11, -12], [-1, -1]).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-affine-recurrence-window · candidate
39. Ordered Recurrence Window and Adjacent Differences
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. a_0=1, a_(n+1)=a_n+(0), start=11, length=3. Give (window indices, term values, adjacent differences); length=0 gives three empty lists.
View answer and explanation
([11, 12, 13], [1, 1, 1], [0, 0])
Apply the declared recurrence from its given initial term and record the ordered window; the index list and repetitions remain meaningful. The exact result, retaining the requested order, is ([11, 12, 13], [1, 1, 1], [0, 0]).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-affine-recurrence-window · candidate
40. Ordered Recurrence Window and Adjacent Differences
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. a_0=2, a_(n+1)=a_n+(1), start=12, length=3. Give (window indices, term values, adjacent differences); length=0 gives three empty lists.
View answer and explanation
([12, 13, 14], [14, 15, 16], [1, 1])
Apply the declared recurrence from its given initial term and record the ordered window; the index list and repetitions remain meaningful. The exact result, retaining the requested order, is ([12, 13, 14], [14, 15, 16], [1, 1]).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-affine-recurrence-window · candidate
41. Ordered Recurrence Window and Adjacent Differences
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. a_0=3, a_(n+1)=a_n+(-1), start=13, length=3. Give (window indices, term values, adjacent differences); length=0 gives three empty lists.
View answer and explanation
([13, 14, 15], [-10, -11, -12], [-1, -1])
Apply the declared recurrence from its given initial term and record the ordered window; the index list and repetitions remain meaningful. The exact result, retaining the requested order, is ([13, 14, 15], [-10, -11, -12], [-1, -1]).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-affine-recurrence-window · candidate
42. Ordered Recurrence Window and Adjacent Differences
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. a_0=4, a_(n+1)=a_n+(0), start=14, length=3. Give (window indices, term values, adjacent differences); length=0 gives three empty lists.
View answer and explanation
([14, 15, 16], [4, 4, 4], [0, 0])
Apply the declared recurrence from its given initial term and record the ordered window; the index list and repetitions remain meaningful. The exact result, retaining the requested order, is ([14, 15, 16], [4, 4, 4], [0, 0]).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-affine-recurrence-window · candidate
43. Ordered Recurrence Window and Adjacent Differences
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. a_0=5, a_(n+1)=a_n+(1), start=15, length=3. Give (window indices, term values, adjacent differences); length=0 gives three empty lists.
View answer and explanation
([15, 16, 17], [20, 21, 22], [1, 1])
Apply the declared recurrence from its given initial term and record the ordered window; the index list and repetitions remain meaningful. The exact result, retaining the requested order, is ([15, 16, 17], [20, 21, 22], [1, 1]).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-affine-recurrence-window · candidate
44. Ordered Recurrence Window and Adjacent Differences
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. a_0=6, a_(n+1)=a_n+(-1), start=16, length=3. Give (window indices, term values, adjacent differences); length=0 gives three empty lists.
View answer and explanation
([16, 17, 18], [-10, -11, -12], [-1, -1])
Apply the declared recurrence from its given initial term and record the ordered window; the index list and repetitions remain meaningful. The exact result, retaining the requested order, is ([16, 17, 18], [-10, -11, -12], [-1, -1]).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-affine-recurrence-window · candidate
45. Ordered Recurrence Window and Adjacent Differences
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. a_0=7, a_(n+1)=a_n+(0), start=17, length=3. Give (window indices, term values, adjacent differences); length=0 gives three empty lists.
View answer and explanation
([17, 18, 19], [7, 7, 7], [0, 0])
Apply the declared recurrence from its given initial term and record the ordered window; the index list and repetitions remain meaningful. The exact result, retaining the requested order, is ([17, 18, 19], [7, 7, 7], [0, 0]).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-affine-recurrence-window · candidate
46. Ordered Recurrence Window and Adjacent Differences
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. a_0=8, a_(n+1)=a_n+(1), start=18, length=3. Give (window indices, term values, adjacent differences); length=0 gives three empty lists.
View answer and explanation
([18, 19, 20], [26, 27, 28], [1, 1])
Apply the declared recurrence from its given initial term and record the ordered window; the index list and repetitions remain meaningful. The exact result, retaining the requested order, is ([18, 19, 20], [26, 27, 28], [1, 1]).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-affine-recurrence-window · candidate
47. Ordered Recurrence Window and Adjacent Differences
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. a_0=9, a_(n+1)=a_n+(-1), start=19, length=3. Give (window indices, term values, adjacent differences); length=0 gives three empty lists.
View answer and explanation
([19, 20, 21], [-10, -11, -12], [-1, -1])
Apply the declared recurrence from its given initial term and record the ordered window; the index list and repetitions remain meaningful. The exact result, retaining the requested order, is ([19, 20, 21], [-10, -11, -12], [-1, -1]).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-affine-recurrence-window · candidate
48. Ordered Recurrence Window and Adjacent Differences
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. a_0=10, a_(n+1)=a_n+(0), start=20, length=3. Give (window indices, term values, adjacent differences); length=0 gives three empty lists.
View answer and explanation
([20, 21, 22], [10, 10, 10], [0, 0])
Apply the declared recurrence from its given initial term and record the ordered window; the index list and repetitions remain meaningful. The exact result, retaining the requested order, is ([20, 21, 22], [10, 10, 10], [0, 0]).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-affine-recurrence-window · candidate
49. Ordered Recurrence Window and Adjacent Differences
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. a_0=11, a_(n+1)=a_n+(1), start=21, length=3. Give (window indices, term values, adjacent differences); length=0 gives three empty lists.
View answer and explanation
([21, 22, 23], [32, 33, 34], [1, 1])
Apply the declared recurrence from its given initial term and record the ordered window; the index list and repetitions remain meaningful. The exact result, retaining the requested order, is ([21, 22, 23], [32, 33, 34], [1, 1]).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-affine-recurrence-window · candidate
50. Ordered Recurrence Window and Adjacent Differences
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. a_0=12, a_(n+1)=a_n+(-1), start=22, length=3. Give (window indices, term values, adjacent differences); length=0 gives three empty lists.
View answer and explanation
([22, 23, 24], [-10, -11, -12], [-1, -1])
Apply the declared recurrence from its given initial term and record the ordered window; the index list and repetitions remain meaningful. The exact result, retaining the requested order, is ([22, 23, 24], [-10, -11, -12], [-1, -1]).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-affine-recurrence-window · candidate
51. Two Fifth-Term Extensions of a Given Four-Term Prefix
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Given ordered prefix=[0, 0, 0, 0] and candidate nexts=[0, 0]. Append each candidate separately and give (first five-term list, second five-term list, are the first four terms equal?, are the fifth terms equal?). No general rule is specified.
View answer and explanation
([0, 0, 0, 0, 0], [0, 0, 0, 0, 0], Yes, Yes)
Append the declared candidate values without inventing a unique continuation rule; equal candidates are not different extensions. The exact result, retaining the requested order, is ([0, 0, 0, 0, 0], [0, 0, 0, 0, 0], Yes, Yes).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-two-finite-extensions · candidate
52. Two Fifth-Term Extensions of a Given Four-Term Prefix
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Given ordered prefix=[1, 1, 2, 2] and candidate nexts=[3, 4]. Append each candidate separately and give (first five-term list, second five-term list, are the first four terms equal?, are the fifth terms equal?). No general rule is specified.
View answer and explanation
([1, 1, 2, 2, 3], [1, 1, 2, 2, 4], Yes, No)
Append the declared candidate values without inventing a unique continuation rule; equal candidates are not different extensions. The exact result, retaining the requested order, is ([1, 1, 2, 2, 3], [1, 1, 2, 2, 4], Yes, No).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-two-finite-extensions · candidate
53. Two Fifth-Term Extensions of a Given Four-Term Prefix
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Given ordered prefix=[-10, 0, -10, 1] and candidate nexts=[2, -1]. Append each candidate separately and give (first five-term list, second five-term list, are the first four terms equal?, are the fifth terms equal?). No general rule is specified.
View answer and explanation
([-10, 0, -10, 1, 2], [-10, 0, -10, 1, -1], Yes, No)
Append the declared candidate values without inventing a unique continuation rule; equal candidates are not different extensions. The exact result, retaining the requested order, is ([-10, 0, -10, 1, 2], [-10, 0, -10, 1, -1], Yes, No).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-two-finite-extensions · candidate
54. Two Fifth-Term Extensions of a Given Four-Term Prefix
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Given ordered prefix=[-9, 0, -9, 1] and candidate nexts=[3, -2]. Append each candidate separately and give (first five-term list, second five-term list, are the first four terms equal?, are the fifth terms equal?). No general rule is specified.
View answer and explanation
([-9, 0, -9, 1, 3], [-9, 0, -9, 1, -2], Yes, No)
Append the declared candidate values without inventing a unique continuation rule; equal candidates are not different extensions. The exact result, retaining the requested order, is ([-9, 0, -9, 1, 3], [-9, 0, -9, 1, -2], Yes, No).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-two-finite-extensions · candidate
55. Two Fifth-Term Extensions of a Given Four-Term Prefix
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Given ordered prefix=[-8, 0, -8, 1] and candidate nexts=[4, -3]. Append each candidate separately and give (first five-term list, second five-term list, are the first four terms equal?, are the fifth terms equal?). No general rule is specified.
View answer and explanation
([-8, 0, -8, 1, 4], [-8, 0, -8, 1, -3], Yes, No)
Append the declared candidate values without inventing a unique continuation rule; equal candidates are not different extensions. The exact result, retaining the requested order, is ([-8, 0, -8, 1, 4], [-8, 0, -8, 1, -3], Yes, No).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-two-finite-extensions · candidate
56. Two Fifth-Term Extensions of a Given Four-Term Prefix
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Given ordered prefix=[-7, 0, -7, 1] and candidate nexts=[5, -4]. Append each candidate separately and give (first five-term list, second five-term list, are the first four terms equal?, are the fifth terms equal?). No general rule is specified.
View answer and explanation
([-7, 0, -7, 1, 5], [-7, 0, -7, 1, -4], Yes, No)
Append the declared candidate values without inventing a unique continuation rule; equal candidates are not different extensions. The exact result, retaining the requested order, is ([-7, 0, -7, 1, 5], [-7, 0, -7, 1, -4], Yes, No).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-two-finite-extensions · candidate
57. Two Fifth-Term Extensions of a Given Four-Term Prefix
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Given ordered prefix=[-6, 0, -6, 1] and candidate nexts=[6, -5]. Append each candidate separately and give (first five-term list, second five-term list, are the first four terms equal?, are the fifth terms equal?). No general rule is specified.
View answer and explanation
([-6, 0, -6, 1, 6], [-6, 0, -6, 1, -5], Yes, No)
Append the declared candidate values without inventing a unique continuation rule; equal candidates are not different extensions. The exact result, retaining the requested order, is ([-6, 0, -6, 1, 6], [-6, 0, -6, 1, -5], Yes, No).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-two-finite-extensions · candidate
58. Two Fifth-Term Extensions of a Given Four-Term Prefix
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Given ordered prefix=[-5, 0, -5, 1] and candidate nexts=[7, -6]. Append each candidate separately and give (first five-term list, second five-term list, are the first four terms equal?, are the fifth terms equal?). No general rule is specified.
View answer and explanation
([-5, 0, -5, 1, 7], [-5, 0, -5, 1, -6], Yes, No)
Append the declared candidate values without inventing a unique continuation rule; equal candidates are not different extensions. The exact result, retaining the requested order, is ([-5, 0, -5, 1, 7], [-5, 0, -5, 1, -6], Yes, No).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-two-finite-extensions · candidate
59. Two Fifth-Term Extensions of a Given Four-Term Prefix
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Given ordered prefix=[-4, 0, -4, 1] and candidate nexts=[8, -7]. Append each candidate separately and give (first five-term list, second five-term list, are the first four terms equal?, are the fifth terms equal?). No general rule is specified.
View answer and explanation
([-4, 0, -4, 1, 8], [-4, 0, -4, 1, -7], Yes, No)
Append the declared candidate values without inventing a unique continuation rule; equal candidates are not different extensions. The exact result, retaining the requested order, is ([-4, 0, -4, 1, 8], [-4, 0, -4, 1, -7], Yes, No).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-two-finite-extensions · candidate
60. Two Fifth-Term Extensions of a Given Four-Term Prefix
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Given ordered prefix=[-3, 0, -3, 1] and candidate nexts=[9, -8]. Append each candidate separately and give (first five-term list, second five-term list, are the first four terms equal?, are the fifth terms equal?). No general rule is specified.
View answer and explanation
([-3, 0, -3, 1, 9], [-3, 0, -3, 1, -8], Yes, No)
Append the declared candidate values without inventing a unique continuation rule; equal candidates are not different extensions. The exact result, retaining the requested order, is ([-3, 0, -3, 1, 9], [-3, 0, -3, 1, -8], Yes, No).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-two-finite-extensions · candidate
61. Two Fifth-Term Extensions of a Given Four-Term Prefix
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Given ordered prefix=[-2, 0, -2, 1] and candidate nexts=[10, -9]. Append each candidate separately and give (first five-term list, second five-term list, are the first four terms equal?, are the fifth terms equal?). No general rule is specified.
View answer and explanation
([-2, 0, -2, 1, 10], [-2, 0, -2, 1, -9], Yes, No)
Append the declared candidate values without inventing a unique continuation rule; equal candidates are not different extensions. The exact result, retaining the requested order, is ([-2, 0, -2, 1, 10], [-2, 0, -2, 1, -9], Yes, No).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-two-finite-extensions · candidate
62. Two Fifth-Term Extensions of a Given Four-Term Prefix
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Given ordered prefix=[-1, 0, -1, 1] and candidate nexts=[11, -10]. Append each candidate separately and give (first five-term list, second five-term list, are the first four terms equal?, are the fifth terms equal?). No general rule is specified.
View answer and explanation
([-1, 0, -1, 1, 11], [-1, 0, -1, 1, -10], Yes, No)
Append the declared candidate values without inventing a unique continuation rule; equal candidates are not different extensions. The exact result, retaining the requested order, is ([-1, 0, -1, 1, 11], [-1, 0, -1, 1, -10], Yes, No).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-two-finite-extensions · candidate
63. Two Fifth-Term Extensions of a Given Four-Term Prefix
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Given ordered prefix=[0, 0, 0, 1] and candidate nexts=[12, -11]. Append each candidate separately and give (first five-term list, second five-term list, are the first four terms equal?, are the fifth terms equal?). No general rule is specified.
View answer and explanation
([0, 0, 0, 1, 12], [0, 0, 0, 1, -11], Yes, No)
Append the declared candidate values without inventing a unique continuation rule; equal candidates are not different extensions. The exact result, retaining the requested order, is ([0, 0, 0, 1, 12], [0, 0, 0, 1, -11], Yes, No).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-two-finite-extensions · candidate
64. Two Fifth-Term Extensions of a Given Four-Term Prefix
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Given ordered prefix=[1, 0, 1, 1] and candidate nexts=[13, -12]. Append each candidate separately and give (first five-term list, second five-term list, are the first four terms equal?, are the fifth terms equal?). No general rule is specified.
View answer and explanation
([1, 0, 1, 1, 13], [1, 0, 1, 1, -12], Yes, No)
Append the declared candidate values without inventing a unique continuation rule; equal candidates are not different extensions. The exact result, retaining the requested order, is ([1, 0, 1, 1, 13], [1, 0, 1, 1, -12], Yes, No).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-two-finite-extensions · candidate
65. Two Fifth-Term Extensions of a Given Four-Term Prefix
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Given ordered prefix=[2, 0, 2, 1] and candidate nexts=[14, -13]. Append each candidate separately and give (first five-term list, second five-term list, are the first four terms equal?, are the fifth terms equal?). No general rule is specified.
View answer and explanation
([2, 0, 2, 1, 14], [2, 0, 2, 1, -13], Yes, No)
Append the declared candidate values without inventing a unique continuation rule; equal candidates are not different extensions. The exact result, retaining the requested order, is ([2, 0, 2, 1, 14], [2, 0, 2, 1, -13], Yes, No).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-two-finite-extensions · candidate
66. Two Fifth-Term Extensions of a Given Four-Term Prefix
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Given ordered prefix=[3, 0, 3, 1] and candidate nexts=[15, -14]. Append each candidate separately and give (first five-term list, second five-term list, are the first four terms equal?, are the fifth terms equal?). No general rule is specified.
View answer and explanation
([3, 0, 3, 1, 15], [3, 0, 3, 1, -14], Yes, No)
Append the declared candidate values without inventing a unique continuation rule; equal candidates are not different extensions. The exact result, retaining the requested order, is ([3, 0, 3, 1, 15], [3, 0, 3, 1, -14], Yes, No).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-two-finite-extensions · candidate
67. Two Fifth-Term Extensions of a Given Four-Term Prefix
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Given ordered prefix=[4, 0, 4, 1] and candidate nexts=[16, -15]. Append each candidate separately and give (first five-term list, second five-term list, are the first four terms equal?, are the fifth terms equal?). No general rule is specified.
View answer and explanation
([4, 0, 4, 1, 16], [4, 0, 4, 1, -15], Yes, No)
Append the declared candidate values without inventing a unique continuation rule; equal candidates are not different extensions. The exact result, retaining the requested order, is ([4, 0, 4, 1, 16], [4, 0, 4, 1, -15], Yes, No).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-two-finite-extensions · candidate
68. Two Fifth-Term Extensions of a Given Four-Term Prefix
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Given ordered prefix=[5, 0, 5, 1] and candidate nexts=[17, -16]. Append each candidate separately and give (first five-term list, second five-term list, are the first four terms equal?, are the fifth terms equal?). No general rule is specified.
View answer and explanation
([5, 0, 5, 1, 17], [5, 0, 5, 1, -16], Yes, No)
Append the declared candidate values without inventing a unique continuation rule; equal candidates are not different extensions. The exact result, retaining the requested order, is ([5, 0, 5, 1, 17], [5, 0, 5, 1, -16], Yes, No).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-two-finite-extensions · candidate
69. Two Fifth-Term Extensions of a Given Four-Term Prefix
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Given ordered prefix=[6, 0, 6, 1] and candidate nexts=[18, -17]. Append each candidate separately and give (first five-term list, second five-term list, are the first four terms equal?, are the fifth terms equal?). No general rule is specified.
View answer and explanation
([6, 0, 6, 1, 18], [6, 0, 6, 1, -17], Yes, No)
Append the declared candidate values without inventing a unique continuation rule; equal candidates are not different extensions. The exact result, retaining the requested order, is ([6, 0, 6, 1, 18], [6, 0, 6, 1, -17], Yes, No).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-two-finite-extensions · candidate
70. Two Fifth-Term Extensions of a Given Four-Term Prefix
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Given ordered prefix=[7, 0, 7, 1] and candidate nexts=[19, -18]. Append each candidate separately and give (first five-term list, second five-term list, are the first four terms equal?, are the fifth terms equal?). No general rule is specified.
View answer and explanation
([7, 0, 7, 1, 19], [7, 0, 7, 1, -18], Yes, No)
Append the declared candidate values without inventing a unique continuation rule; equal candidates are not different extensions. The exact result, retaining the requested order, is ([7, 0, 7, 1, 19], [7, 0, 7, 1, -18], Yes, No).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-two-finite-extensions · candidate
71. Two Fifth-Term Extensions of a Given Four-Term Prefix
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Given ordered prefix=[8, 0, 8, 1] and candidate nexts=[20, -19]. Append each candidate separately and give (first five-term list, second five-term list, are the first four terms equal?, are the fifth terms equal?). No general rule is specified.
View answer and explanation
([8, 0, 8, 1, 20], [8, 0, 8, 1, -19], Yes, No)
Append the declared candidate values without inventing a unique continuation rule; equal candidates are not different extensions. The exact result, retaining the requested order, is ([8, 0, 8, 1, 20], [8, 0, 8, 1, -19], Yes, No).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-two-finite-extensions · candidate
72. Two Fifth-Term Extensions of a Given Four-Term Prefix
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Given ordered prefix=[9, 0, 9, 1] and candidate nexts=[21, -20]. Append each candidate separately and give (first five-term list, second five-term list, are the first four terms equal?, are the fifth terms equal?). No general rule is specified.
View answer and explanation
([9, 0, 9, 1, 21], [9, 0, 9, 1, -20], Yes, No)
Append the declared candidate values without inventing a unique continuation rule; equal candidates are not different extensions. The exact result, retaining the requested order, is ([9, 0, 9, 1, 21], [9, 0, 9, 1, -20], Yes, No).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-two-finite-extensions · candidate
73. Two Fifth-Term Extensions of a Given Four-Term Prefix
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Given ordered prefix=[10, 0, 10, 1] and candidate nexts=[22, -21]. Append each candidate separately and give (first five-term list, second five-term list, are the first four terms equal?, are the fifth terms equal?). No general rule is specified.
View answer and explanation
([10, 0, 10, 1, 22], [10, 0, 10, 1, -21], Yes, No)
Append the declared candidate values without inventing a unique continuation rule; equal candidates are not different extensions. The exact result, retaining the requested order, is ([10, 0, 10, 1, 22], [10, 0, 10, 1, -21], Yes, No).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-two-finite-extensions · candidate
74. Two Fifth-Term Extensions of a Given Four-Term Prefix
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Given ordered prefix=[11, 0, 11, 1] and candidate nexts=[23, -22]. Append each candidate separately and give (first five-term list, second five-term list, are the first four terms equal?, are the fifth terms equal?). No general rule is specified.
View answer and explanation
([11, 0, 11, 1, 23], [11, 0, 11, 1, -22], Yes, No)
Append the declared candidate values without inventing a unique continuation rule; equal candidates are not different extensions. The exact result, retaining the requested order, is ([11, 0, 11, 1, 23], [11, 0, 11, 1, -22], Yes, No).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-two-finite-extensions · candidate
75. Two Fifth-Term Extensions of a Given Four-Term Prefix
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Given ordered prefix=[12, 0, 12, 1] and candidate nexts=[24, -23]. Append each candidate separately and give (first five-term list, second five-term list, are the first four terms equal?, are the fifth terms equal?). No general rule is specified.
View answer and explanation
([12, 0, 12, 1, 24], [12, 0, 12, 1, -23], Yes, No)
Append the declared candidate values without inventing a unique continuation rule; equal candidates are not different extensions. The exact result, retaining the requested order, is ([12, 0, 12, 1, 24], [12, 0, 12, 1, -23], Yes, No).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-two-finite-extensions · candidate
76. Five Prefix Sums and Four Ordered Differences
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Five ordered values=[0, 0, 0, 0, 0]. Give (five inclusive prefix sums, four following-minus-previous differences, are all five values equal?).
View answer and explanation
([0, 0, 0, 0, 0], [0, 0, 0, 0], Yes)
Accumulate in order and subtract adjacent values, without sorting or deduplicating; finite sums are not an infinite series or limit. The exact result, retaining the requested order, is ([0, 0, 0, 0, 0], [0, 0, 0, 0], Yes).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-five-term-prefix-deltas · candidate
77. Five Prefix Sums and Four Ordered Differences
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Five ordered values=[2, 2, 2, 2, 2]. Give (five inclusive prefix sums, four following-minus-previous differences, are all five values equal?).
View answer and explanation
([2, 4, 6, 8, 10], [0, 0, 0, 0], Yes)
Accumulate in order and subtract adjacent values, without sorting or deduplicating; finite sums are not an infinite series or limit. The exact result, retaining the requested order, is ([2, 4, 6, 8, 10], [0, 0, 0, 0], Yes).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-five-term-prefix-deltas · candidate
78. Five Prefix Sums and Four Ordered Differences
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Five ordered values=[-2, -1, 0, 1, 2]. Give (five inclusive prefix sums, four following-minus-previous differences, are all five values equal?).
View answer and explanation
([-2, -3, -3, -2, 0], [1, 1, 1, 1], No)
Accumulate in order and subtract adjacent values, without sorting or deduplicating; finite sums are not an infinite series or limit. The exact result, retaining the requested order, is ([-2, -3, -3, -2, 0], [1, 1, 1, 1], No).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-five-term-prefix-deltas · candidate
79. Five Prefix Sums and Four Ordered Differences
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Five ordered values=[-9, 0, -9, -1, 1]. Give (five inclusive prefix sums, four following-minus-previous differences, are all five values equal?).
View answer and explanation
([-9, -9, -18, -19, -18], [9, -9, 8, 2], No)
Accumulate in order and subtract adjacent values, without sorting or deduplicating; finite sums are not an infinite series or limit. The exact result, retaining the requested order, is ([-9, -9, -18, -19, -18], [9, -9, 8, 2], No).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-five-term-prefix-deltas · candidate
80. Five Prefix Sums and Four Ordered Differences
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Five ordered values=[-8, 0, -8, -1, 1]. Give (five inclusive prefix sums, four following-minus-previous differences, are all five values equal?).
View answer and explanation
([-8, -8, -16, -17, -16], [8, -8, 7, 2], No)
Accumulate in order and subtract adjacent values, without sorting or deduplicating; finite sums are not an infinite series or limit. The exact result, retaining the requested order, is ([-8, -8, -16, -17, -16], [8, -8, 7, 2], No).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-five-term-prefix-deltas · candidate
81. Five Prefix Sums and Four Ordered Differences
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Five ordered values=[-7, 0, -7, -1, 1]. Give (five inclusive prefix sums, four following-minus-previous differences, are all five values equal?).
View answer and explanation
([-7, -7, -14, -15, -14], [7, -7, 6, 2], No)
Accumulate in order and subtract adjacent values, without sorting or deduplicating; finite sums are not an infinite series or limit. The exact result, retaining the requested order, is ([-7, -7, -14, -15, -14], [7, -7, 6, 2], No).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-five-term-prefix-deltas · candidate
82. Five Prefix Sums and Four Ordered Differences
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Five ordered values=[-6, 0, -6, -1, 1]. Give (five inclusive prefix sums, four following-minus-previous differences, are all five values equal?).
View answer and explanation
([-6, -6, -12, -13, -12], [6, -6, 5, 2], No)
Accumulate in order and subtract adjacent values, without sorting or deduplicating; finite sums are not an infinite series or limit. The exact result, retaining the requested order, is ([-6, -6, -12, -13, -12], [6, -6, 5, 2], No).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-five-term-prefix-deltas · candidate
83. Five Prefix Sums and Four Ordered Differences
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Five ordered values=[-5, 0, -5, -1, 1]. Give (five inclusive prefix sums, four following-minus-previous differences, are all five values equal?).
View answer and explanation
([-5, -5, -10, -11, -10], [5, -5, 4, 2], No)
Accumulate in order and subtract adjacent values, without sorting or deduplicating; finite sums are not an infinite series or limit. The exact result, retaining the requested order, is ([-5, -5, -10, -11, -10], [5, -5, 4, 2], No).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-five-term-prefix-deltas · candidate
84. Five Prefix Sums and Four Ordered Differences
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Five ordered values=[-4, 0, -4, -1, 1]. Give (five inclusive prefix sums, four following-minus-previous differences, are all five values equal?).
View answer and explanation
([-4, -4, -8, -9, -8], [4, -4, 3, 2], No)
Accumulate in order and subtract adjacent values, without sorting or deduplicating; finite sums are not an infinite series or limit. The exact result, retaining the requested order, is ([-4, -4, -8, -9, -8], [4, -4, 3, 2], No).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-five-term-prefix-deltas · candidate
85. Five Prefix Sums and Four Ordered Differences
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Five ordered values=[-3, 0, -3, -1, 1]. Give (five inclusive prefix sums, four following-minus-previous differences, are all five values equal?).
View answer and explanation
([-3, -3, -6, -7, -6], [3, -3, 2, 2], No)
Accumulate in order and subtract adjacent values, without sorting or deduplicating; finite sums are not an infinite series or limit. The exact result, retaining the requested order, is ([-3, -3, -6, -7, -6], [3, -3, 2, 2], No).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-five-term-prefix-deltas · candidate
86. Five Prefix Sums and Four Ordered Differences
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Five ordered values=[-2, 0, -2, -1, 1]. Give (five inclusive prefix sums, four following-minus-previous differences, are all five values equal?).
View answer and explanation
([-2, -2, -4, -5, -4], [2, -2, 1, 2], No)
Accumulate in order and subtract adjacent values, without sorting or deduplicating; finite sums are not an infinite series or limit. The exact result, retaining the requested order, is ([-2, -2, -4, -5, -4], [2, -2, 1, 2], No).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-five-term-prefix-deltas · candidate
87. Five Prefix Sums and Four Ordered Differences
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Five ordered values=[-1, 0, -1, -1, 1]. Give (five inclusive prefix sums, four following-minus-previous differences, are all five values equal?).
View answer and explanation
([-1, -1, -2, -3, -2], [1, -1, 0, 2], No)
Accumulate in order and subtract adjacent values, without sorting or deduplicating; finite sums are not an infinite series or limit. The exact result, retaining the requested order, is ([-1, -1, -2, -3, -2], [1, -1, 0, 2], No).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-five-term-prefix-deltas · candidate
88. Five Prefix Sums and Four Ordered Differences
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Five ordered values=[0, 0, 0, -1, 1]. Give (five inclusive prefix sums, four following-minus-previous differences, are all five values equal?).
View answer and explanation
([0, 0, 0, -1, 0], [0, 0, -1, 2], No)
Accumulate in order and subtract adjacent values, without sorting or deduplicating; finite sums are not an infinite series or limit. The exact result, retaining the requested order, is ([0, 0, 0, -1, 0], [0, 0, -1, 2], No).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-five-term-prefix-deltas · candidate
89. Five Prefix Sums and Four Ordered Differences
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Five ordered values=[1, 0, 1, -1, 1]. Give (five inclusive prefix sums, four following-minus-previous differences, are all five values equal?).
View answer and explanation
([1, 1, 2, 1, 2], [-1, 1, -2, 2], No)
Accumulate in order and subtract adjacent values, without sorting or deduplicating; finite sums are not an infinite series or limit. The exact result, retaining the requested order, is ([1, 1, 2, 1, 2], [-1, 1, -2, 2], No).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-five-term-prefix-deltas · candidate
90. Five Prefix Sums and Four Ordered Differences
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Five ordered values=[2, 0, 2, -1, 1]. Give (five inclusive prefix sums, four following-minus-previous differences, are all five values equal?).
View answer and explanation
([2, 2, 4, 3, 4], [-2, 2, -3, 2], No)
Accumulate in order and subtract adjacent values, without sorting or deduplicating; finite sums are not an infinite series or limit. The exact result, retaining the requested order, is ([2, 2, 4, 3, 4], [-2, 2, -3, 2], No).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-five-term-prefix-deltas · candidate
91. Five Prefix Sums and Four Ordered Differences
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Five ordered values=[3, 0, 3, -1, 1]. Give (five inclusive prefix sums, four following-minus-previous differences, are all five values equal?).
View answer and explanation
([3, 3, 6, 5, 6], [-3, 3, -4, 2], No)
Accumulate in order and subtract adjacent values, without sorting or deduplicating; finite sums are not an infinite series or limit. The exact result, retaining the requested order, is ([3, 3, 6, 5, 6], [-3, 3, -4, 2], No).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-five-term-prefix-deltas · candidate
92. Five Prefix Sums and Four Ordered Differences
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Five ordered values=[4, 0, 4, -1, 1]. Give (five inclusive prefix sums, four following-minus-previous differences, are all five values equal?).
View answer and explanation
([4, 4, 8, 7, 8], [-4, 4, -5, 2], No)
Accumulate in order and subtract adjacent values, without sorting or deduplicating; finite sums are not an infinite series or limit. The exact result, retaining the requested order, is ([4, 4, 8, 7, 8], [-4, 4, -5, 2], No).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-five-term-prefix-deltas · candidate
93. Five Prefix Sums and Four Ordered Differences
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Five ordered values=[5, 0, 5, -1, 1]. Give (five inclusive prefix sums, four following-minus-previous differences, are all five values equal?).
View answer and explanation
([5, 5, 10, 9, 10], [-5, 5, -6, 2], No)
Accumulate in order and subtract adjacent values, without sorting or deduplicating; finite sums are not an infinite series or limit. The exact result, retaining the requested order, is ([5, 5, 10, 9, 10], [-5, 5, -6, 2], No).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-five-term-prefix-deltas · candidate
94. Five Prefix Sums and Four Ordered Differences
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Five ordered values=[6, 0, 6, -1, 1]. Give (five inclusive prefix sums, four following-minus-previous differences, are all five values equal?).
View answer and explanation
([6, 6, 12, 11, 12], [-6, 6, -7, 2], No)
Accumulate in order and subtract adjacent values, without sorting or deduplicating; finite sums are not an infinite series or limit. The exact result, retaining the requested order, is ([6, 6, 12, 11, 12], [-6, 6, -7, 2], No).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-five-term-prefix-deltas · candidate
95. Five Prefix Sums and Four Ordered Differences
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Five ordered values=[7, 0, 7, -1, 1]. Give (five inclusive prefix sums, four following-minus-previous differences, are all five values equal?).
View answer and explanation
([7, 7, 14, 13, 14], [-7, 7, -8, 2], No)
Accumulate in order and subtract adjacent values, without sorting or deduplicating; finite sums are not an infinite series or limit. The exact result, retaining the requested order, is ([7, 7, 14, 13, 14], [-7, 7, -8, 2], No).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-five-term-prefix-deltas · candidate
96. Five Prefix Sums and Four Ordered Differences
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Five ordered values=[8, 0, 8, -1, 1]. Give (five inclusive prefix sums, four following-minus-previous differences, are all five values equal?).
View answer and explanation
([8, 8, 16, 15, 16], [-8, 8, -9, 2], No)
Accumulate in order and subtract adjacent values, without sorting or deduplicating; finite sums are not an infinite series or limit. The exact result, retaining the requested order, is ([8, 8, 16, 15, 16], [-8, 8, -9, 2], No).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-five-term-prefix-deltas · candidate
97. Five Prefix Sums and Four Ordered Differences
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Five ordered values=[9, 0, 9, -1, 1]. Give (five inclusive prefix sums, four following-minus-previous differences, are all five values equal?).
View answer and explanation
([9, 9, 18, 17, 18], [-9, 9, -10, 2], No)
Accumulate in order and subtract adjacent values, without sorting or deduplicating; finite sums are not an infinite series or limit. The exact result, retaining the requested order, is ([9, 9, 18, 17, 18], [-9, 9, -10, 2], No).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-five-term-prefix-deltas · candidate
98. Five Prefix Sums and Four Ordered Differences
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Five ordered values=[10, 0, 10, -1, 1]. Give (five inclusive prefix sums, four following-minus-previous differences, are all five values equal?).
View answer and explanation
([10, 10, 20, 19, 20], [-10, 10, -11, 2], No)
Accumulate in order and subtract adjacent values, without sorting or deduplicating; finite sums are not an infinite series or limit. The exact result, retaining the requested order, is ([10, 10, 20, 19, 20], [-10, 10, -11, 2], No).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-five-term-prefix-deltas · candidate
99. Five Prefix Sums and Four Ordered Differences
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Five ordered values=[11, 0, 11, -1, 1]. Give (five inclusive prefix sums, four following-minus-previous differences, are all five values equal?).
View answer and explanation
([11, 11, 22, 21, 22], [-11, 11, -12, 2], No)
Accumulate in order and subtract adjacent values, without sorting or deduplicating; finite sums are not an infinite series or limit. The exact result, retaining the requested order, is ([11, 11, 22, 21, 22], [-11, 11, -12, 2], No).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-five-term-prefix-deltas · candidate
100. Five Prefix Sums and Four Ordered Differences
N begins at 0; retain sequence order and repeated values. A finite prefix does not determine a unique infinite rule, convergence, or infinite series. Five ordered values=[12, 0, 12, -1, 1]. Give (five inclusive prefix sums, four following-minus-previous differences, are all five values equal?).
View answer and explanation
([12, 12, 24, 23, 24], [-12, 12, -13, 2], No)
Accumulate in order and subtract adjacent values, without sorting or deduplicating; finite sums are not an infinite series or limit. The exact result, retaining the requested order, is ([12, 12, 24, 23, 24], [-12, 12, -13, 2], No).
Reviewed step-by-step derivation
Sequence · qf-density15-sequence-five-term-prefix-deltas · candidate
Canonical question JSONL · English JSONL · Validation report