This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.
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-20260930-r05-sequence-foundations · 2026-09-30 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
The real sequence is indexed from 0 and has the explicit rule a_n=(-12)+(-3)n for n∈N={0,1,2,…}. Find the term at n=0.
-12
Substitute the explicit zero-based index: a_0=-12+(-3)×0=-12. No limit claim is involved.
Reviewed step-by-step derivation
Sequence · qf-sequence-explicit-term · candidate
The real sequence is indexed from 0 and has the explicit rule a_n=(-11)+(-2)n for n∈N={0,1,2,…}. Find the term at n=1.
-13
Substitute the explicit zero-based index: a_1=-11+(-2)×1=-13. No limit claim is involved.
Reviewed step-by-step derivation
Sequence · qf-sequence-explicit-term · candidate
The real sequence is indexed from 0 and has the explicit rule a_n=(-10)+(-1)n for n∈N={0,1,2,…}. Find the term at n=2.
-12
Substitute the explicit zero-based index: a_2=-10+(-1)×2=-12. No limit claim is involved.
Reviewed step-by-step derivation
Sequence · qf-sequence-explicit-term · candidate
The real sequence is indexed from 0 and has the explicit rule a_n=(-9)+(0)n for n∈N={0,1,2,…}. Find the term at n=3.
-9
Substitute the explicit zero-based index: a_3=-9+(0)×3=-9. No limit claim is involved.
Reviewed step-by-step derivation
Sequence · qf-sequence-explicit-term · candidate
The real sequence is indexed from 0 and has the explicit rule a_n=(-8)+(1)n for n∈N={0,1,2,…}. Find the term at n=4.
-4
Substitute the explicit zero-based index: a_4=-8+(1)×4=-4. No limit claim is involved.
Reviewed step-by-step derivation
Sequence · qf-sequence-explicit-term · candidate
The real sequence is indexed from 0 and has the explicit rule a_n=(-7)+(2)n for n∈N={0,1,2,…}. Find the term at n=5.
3
Substitute the explicit zero-based index: a_5=-7+(2)×5=3. No limit claim is involved.
Reviewed step-by-step derivation
Sequence · qf-sequence-explicit-term · candidate
The real sequence is indexed from 0 and has the explicit rule a_n=(-6)+(3)n for n∈N={0,1,2,…}. Find the term at n=6.
12
Substitute the explicit zero-based index: a_6=-6+(3)×6=12. No limit claim is involved.
Reviewed step-by-step derivation
Sequence · qf-sequence-explicit-term · candidate
The real sequence is indexed from 0 and has the explicit rule a_n=(-5)+(-3)n for n∈N={0,1,2,…}. Find the term at n=7.
-26
Substitute the explicit zero-based index: a_7=-5+(-3)×7=-26. No limit claim is involved.
Reviewed step-by-step derivation
Sequence · qf-sequence-explicit-term · candidate
The real sequence is indexed from 0 and has the explicit rule a_n=(-4)+(-2)n for n∈N={0,1,2,…}. Find the term at n=8.
-20
Substitute the explicit zero-based index: a_8=-4+(-2)×8=-20. No limit claim is involved.
Reviewed step-by-step derivation
Sequence · qf-sequence-explicit-term · candidate
The real sequence is indexed from 0 and has the explicit rule a_n=(-3)+(-1)n for n∈N={0,1,2,…}. Find the term at n=9.
-12
Substitute the explicit zero-based index: a_9=-3+(-1)×9=-12. No limit claim is involved.
Reviewed step-by-step derivation
Sequence · qf-sequence-explicit-term · candidate
The real sequence is indexed from 0 and has the explicit rule a_n=(-2)+(0)n for n∈N={0,1,2,…}. Find the term at n=0.
-2
Substitute the explicit zero-based index: a_0=-2+(0)×0=-2. No limit claim is involved.
Reviewed step-by-step derivation
Sequence · qf-sequence-explicit-term · candidate
The real sequence is indexed from 0 and has the explicit rule a_n=(-1)+(1)n for n∈N={0,1,2,…}. Find the term at n=1.
0
Substitute the explicit zero-based index: a_1=-1+(1)×1=0. No limit claim is involved.
Reviewed step-by-step derivation
Sequence · qf-sequence-explicit-term · candidate
The real sequence is indexed from 0 and has the explicit rule a_n=(0)+(2)n for n∈N={0,1,2,…}. Find the term at n=2.
4
Substitute the explicit zero-based index: a_2=0+(2)×2=4. No limit claim is involved.
Reviewed step-by-step derivation
Sequence · qf-sequence-explicit-term · candidate
The real sequence is indexed from 0 and has the explicit rule a_n=(1)+(3)n for n∈N={0,1,2,…}. Find the term at n=3.
10
Substitute the explicit zero-based index: a_3=1+(3)×3=10. No limit claim is involved.
Reviewed step-by-step derivation
Sequence · qf-sequence-explicit-term · candidate
The real sequence is indexed from 0 and has the explicit rule a_n=(2)+(-3)n for n∈N={0,1,2,…}. Find the term at n=4.
-10
Substitute the explicit zero-based index: a_4=2+(-3)×4=-10. No limit claim is involved.
Reviewed step-by-step derivation
Sequence · qf-sequence-explicit-term · candidate
The real sequence is indexed from 0 and has the explicit rule a_n=(3)+(-2)n for n∈N={0,1,2,…}. Find the term at n=5.
-7
Substitute the explicit zero-based index: a_5=3+(-2)×5=-7. No limit claim is involved.
Reviewed step-by-step derivation
Sequence · qf-sequence-explicit-term · candidate
The real sequence is indexed from 0 and has the explicit rule a_n=(4)+(-1)n for n∈N={0,1,2,…}. Find the term at n=6.
-2
Substitute the explicit zero-based index: a_6=4+(-1)×6=-2. No limit claim is involved.
Reviewed step-by-step derivation
Sequence · qf-sequence-explicit-term · candidate
The real sequence is indexed from 0 and has the explicit rule a_n=(5)+(0)n for n∈N={0,1,2,…}. Find the term at n=7.
5
Substitute the explicit zero-based index: a_7=5+(0)×7=5. No limit claim is involved.
Reviewed step-by-step derivation
Sequence · qf-sequence-explicit-term · candidate
The real sequence is indexed from 0 and has the explicit rule a_n=(6)+(1)n for n∈N={0,1,2,…}. Find the term at n=8.
14
Substitute the explicit zero-based index: a_8=6+(1)×8=14. No limit claim is involved.
Reviewed step-by-step derivation
Sequence · qf-sequence-explicit-term · candidate
The real sequence is indexed from 0 and has the explicit rule a_n=(7)+(2)n for n∈N={0,1,2,…}. Find the term at n=9.
25
Substitute the explicit zero-based index: a_9=7+(2)×9=25. No limit claim is involved.
Reviewed step-by-step derivation
Sequence · qf-sequence-explicit-term · candidate
The real sequence is indexed from 0 and has the explicit rule a_n=(8)+(3)n for n∈N={0,1,2,…}. Find the term at n=0.
8
Substitute the explicit zero-based index: a_0=8+(3)×0=8. No limit claim is involved.
Reviewed step-by-step derivation
Sequence · qf-sequence-explicit-term · candidate
The real sequence is indexed from 0 and has the explicit rule a_n=(9)+(-3)n for n∈N={0,1,2,…}. Find the term at n=1.
6
Substitute the explicit zero-based index: a_1=9+(-3)×1=6. No limit claim is involved.
Reviewed step-by-step derivation
Sequence · qf-sequence-explicit-term · candidate
The real sequence is indexed from 0 and has the explicit rule a_n=(10)+(-2)n for n∈N={0,1,2,…}. Find the term at n=2.
6
Substitute the explicit zero-based index: a_2=10+(-2)×2=6. No limit claim is involved.
Reviewed step-by-step derivation
Sequence · qf-sequence-explicit-term · candidate
The real sequence is indexed from 0 and has the explicit rule a_n=(11)+(-1)n for n∈N={0,1,2,…}. Find the term at n=3.
8
Substitute the explicit zero-based index: a_3=11+(-1)×3=8. No limit claim is involved.
Reviewed step-by-step derivation
Sequence · qf-sequence-explicit-term · candidate
The real sequence is indexed from 0 and has the explicit rule a_n=(12)+(0)n for n∈N={0,1,2,…}. Find the term at n=4.
12
Substitute the explicit zero-based index: a_4=12+(0)×4=12. No limit claim is involved.
Reviewed step-by-step derivation
Sequence · qf-sequence-explicit-term · candidate
The real sequence is indexed from 0 with a_0=12 and recurrence a_(j+1)=a_j+(-3) for every j≥0. Find the term at n=0.
12
Start from a_0 and perform exactly 0 recurrence update(s), obtaining a_0=12.
Reviewed step-by-step derivation
Sequence · qf-sequence-recursive-term · candidate
The real sequence is indexed from 0 with a_0=11 and recurrence a_(j+1)=a_j+(-2) for every j≥0. Find the term at n=3.
5
Start from a_0 and perform exactly 3 recurrence update(s), obtaining a_3=5.
Reviewed step-by-step derivation
Sequence · qf-sequence-recursive-term · candidate
The real sequence is indexed from 0 with a_0=10 and recurrence a_(j+1)=a_j+(-1) for every j≥0. Find the term at n=6.
4
Start from a_0 and perform exactly 6 recurrence update(s), obtaining a_6=4.
Reviewed step-by-step derivation
Sequence · qf-sequence-recursive-term · candidate
The real sequence is indexed from 0 with a_0=9 and recurrence a_(j+1)=a_j+(0) for every j≥0. Find the term at n=9.
9
Start from a_0 and perform exactly 9 recurrence update(s), obtaining a_9=9.
Reviewed step-by-step derivation
Sequence · qf-sequence-recursive-term · candidate
The real sequence is indexed from 0 with a_0=8 and recurrence a_(j+1)=a_j+(1) for every j≥0. Find the term at n=12.
20
Start from a_0 and perform exactly 12 recurrence update(s), obtaining a_12=20.
Reviewed step-by-step derivation
Sequence · qf-sequence-recursive-term · candidate
The real sequence is indexed from 0 with a_0=7 and recurrence a_(j+1)=a_j+(2) for every j≥0. Find the term at n=2.
11
Start from a_0 and perform exactly 2 recurrence update(s), obtaining a_2=11.
Reviewed step-by-step derivation
Sequence · qf-sequence-recursive-term · candidate
The real sequence is indexed from 0 with a_0=6 and recurrence a_(j+1)=a_j+(3) for every j≥0. Find the term at n=5.
21
Start from a_0 and perform exactly 5 recurrence update(s), obtaining a_5=21.
Reviewed step-by-step derivation
Sequence · qf-sequence-recursive-term · candidate
The real sequence is indexed from 0 with a_0=5 and recurrence a_(j+1)=a_j+(-3) for every j≥0. Find the term at n=8.
-19
Start from a_0 and perform exactly 8 recurrence update(s), obtaining a_8=-19.
Reviewed step-by-step derivation
Sequence · qf-sequence-recursive-term · candidate
The real sequence is indexed from 0 with a_0=4 and recurrence a_(j+1)=a_j+(-2) for every j≥0. Find the term at n=11.
-18
Start from a_0 and perform exactly 11 recurrence update(s), obtaining a_11=-18.
Reviewed step-by-step derivation
Sequence · qf-sequence-recursive-term · candidate
The real sequence is indexed from 0 with a_0=3 and recurrence a_(j+1)=a_j+(-1) for every j≥0. Find the term at n=1.
2
Start from a_0 and perform exactly 1 recurrence update(s), obtaining a_1=2.
Reviewed step-by-step derivation
Sequence · qf-sequence-recursive-term · candidate
The real sequence is indexed from 0 with a_0=2 and recurrence a_(j+1)=a_j+(0) for every j≥0. Find the term at n=4.
2
Start from a_0 and perform exactly 4 recurrence update(s), obtaining a_4=2.
Reviewed step-by-step derivation
Sequence · qf-sequence-recursive-term · candidate
The real sequence is indexed from 0 with a_0=1 and recurrence a_(j+1)=a_j+(1) for every j≥0. Find the term at n=7.
8
Start from a_0 and perform exactly 7 recurrence update(s), obtaining a_7=8.
Reviewed step-by-step derivation
Sequence · qf-sequence-recursive-term · candidate
The real sequence is indexed from 0 with a_0=0 and recurrence a_(j+1)=a_j+(2) for every j≥0. Find the term at n=10.
20
Start from a_0 and perform exactly 10 recurrence update(s), obtaining a_10=20.
Reviewed step-by-step derivation
Sequence · qf-sequence-recursive-term · candidate
The real sequence is indexed from 0 with a_0=-1 and recurrence a_(j+1)=a_j+(3) for every j≥0. Find the term at n=0.
-1
Start from a_0 and perform exactly 0 recurrence update(s), obtaining a_0=-1.
Reviewed step-by-step derivation
Sequence · qf-sequence-recursive-term · candidate
The real sequence is indexed from 0 with a_0=-2 and recurrence a_(j+1)=a_j+(-3) for every j≥0. Find the term at n=3.
-11
Start from a_0 and perform exactly 3 recurrence update(s), obtaining a_3=-11.
Reviewed step-by-step derivation
Sequence · qf-sequence-recursive-term · candidate
The real sequence is indexed from 0 with a_0=-3 and recurrence a_(j+1)=a_j+(-2) for every j≥0. Find the term at n=6.
-15
Start from a_0 and perform exactly 6 recurrence update(s), obtaining a_6=-15.
Reviewed step-by-step derivation
Sequence · qf-sequence-recursive-term · candidate
The real sequence is indexed from 0 with a_0=-4 and recurrence a_(j+1)=a_j+(-1) for every j≥0. Find the term at n=9.
-13
Start from a_0 and perform exactly 9 recurrence update(s), obtaining a_9=-13.
Reviewed step-by-step derivation
Sequence · qf-sequence-recursive-term · candidate
The real sequence is indexed from 0 with a_0=-5 and recurrence a_(j+1)=a_j+(0) for every j≥0. Find the term at n=12.
-5
Start from a_0 and perform exactly 12 recurrence update(s), obtaining a_12=-5.
Reviewed step-by-step derivation
Sequence · qf-sequence-recursive-term · candidate
The real sequence is indexed from 0 with a_0=-6 and recurrence a_(j+1)=a_j+(1) for every j≥0. Find the term at n=2.
-4
Start from a_0 and perform exactly 2 recurrence update(s), obtaining a_2=-4.
Reviewed step-by-step derivation
Sequence · qf-sequence-recursive-term · candidate
The real sequence is indexed from 0 with a_0=-7 and recurrence a_(j+1)=a_j+(2) for every j≥0. Find the term at n=5.
3
Start from a_0 and perform exactly 5 recurrence update(s), obtaining a_5=3.
Reviewed step-by-step derivation
Sequence · qf-sequence-recursive-term · candidate
The real sequence is indexed from 0 with a_0=-8 and recurrence a_(j+1)=a_j+(3) for every j≥0. Find the term at n=8.
16
Start from a_0 and perform exactly 8 recurrence update(s), obtaining a_8=16.
Reviewed step-by-step derivation
Sequence · qf-sequence-recursive-term · candidate
The real sequence is indexed from 0 with a_0=-9 and recurrence a_(j+1)=a_j+(-3) for every j≥0. Find the term at n=11.
-42
Start from a_0 and perform exactly 11 recurrence update(s), obtaining a_11=-42.
Reviewed step-by-step derivation
Sequence · qf-sequence-recursive-term · candidate
The real sequence is indexed from 0 with a_0=-10 and recurrence a_(j+1)=a_j+(-2) for every j≥0. Find the term at n=1.
-12
Start from a_0 and perform exactly 1 recurrence update(s), obtaining a_1=-12.
Reviewed step-by-step derivation
Sequence · qf-sequence-recursive-term · candidate
The real sequence is indexed from 0 with a_0=-11 and recurrence a_(j+1)=a_j+(-1) for every j≥0. Find the term at n=4.
-15
Start from a_0 and perform exactly 4 recurrence update(s), obtaining a_4=-15.
Reviewed step-by-step derivation
Sequence · qf-sequence-recursive-term · candidate
The real sequence is indexed from 0 with a_0=-12 and recurrence a_(j+1)=a_j+(0) for every j≥0. Find the term at n=7.
-12
Start from a_0 and perform exactly 7 recurrence update(s), obtaining a_7=-12.
Reviewed step-by-step derivation
Sequence · qf-sequence-recursive-term · candidate
The real sequence is indexed from 0 with a_n=(-12)+(-3)n. List the prefix [a_0,…,a_(L-1)] of length L=0; use [] when L=0.
[]
Take exactly 0 term(s) in index order beginning at 0. The explicit-formula prefix is [], not a guessed continuation rule.
Reviewed step-by-step derivation
Sequence · qf-sequence-prefix · candidate
The real sequence is indexed from 0 with a_n=(-11)+(-2)n. List the prefix [a_0,…,a_(L-1)] of length L=2; use [] when L=0.
[-11, -13]
Take exactly 2 term(s) in index order beginning at 0. The explicit-formula prefix is [-11, -13], not a guessed continuation rule.
Reviewed step-by-step derivation
Sequence · qf-sequence-prefix · candidate
The real sequence is indexed from 0 with a_n=(-10)+(-1)n. List the prefix [a_0,…,a_(L-1)] of length L=3; use [] when L=0.
[-10, -11, -12]
Take exactly 3 term(s) in index order beginning at 0. The explicit-formula prefix is [-10, -11, -12], not a guessed continuation rule.
Reviewed step-by-step derivation
Sequence · qf-sequence-prefix · candidate
The real sequence is indexed from 0 with a_n=(-9)+(0)n. List the prefix [a_0,…,a_(L-1)] of length L=4; use [] when L=0.
[-9, -9, -9, -9]
Take exactly 4 term(s) in index order beginning at 0. The explicit-formula prefix is [-9, -9, -9, -9], not a guessed continuation rule.
Reviewed step-by-step derivation
Sequence · qf-sequence-prefix · candidate
The real sequence is indexed from 0 with a_n=(-8)+(1)n. List the prefix [a_0,…,a_(L-1)] of length L=5; use [] when L=0.
[-8, -7, -6, -5, -4]
Take exactly 5 term(s) in index order beginning at 0. The explicit-formula prefix is [-8, -7, -6, -5, -4], not a guessed continuation rule.
Reviewed step-by-step derivation
Sequence · qf-sequence-prefix · candidate
The real sequence is indexed from 0 with a_n=(-7)+(2)n. List the prefix [a_0,…,a_(L-1)] of length L=6; use [] when L=0.
[-7, -5, -3, -1, 1, 3]
Take exactly 6 term(s) in index order beginning at 0. The explicit-formula prefix is [-7, -5, -3, -1, 1, 3], not a guessed continuation rule.
Reviewed step-by-step derivation
Sequence · qf-sequence-prefix · candidate
The real sequence is indexed from 0 with a_n=(-6)+(3)n. List the prefix [a_0,…,a_(L-1)] of length L=7; use [] when L=0.
[-6, -3, 0, 3, 6, 9, 12]
Take exactly 7 term(s) in index order beginning at 0. The explicit-formula prefix is [-6, -3, 0, 3, 6, 9, 12], not a guessed continuation rule.
Reviewed step-by-step derivation
Sequence · qf-sequence-prefix · candidate
The real sequence is indexed from 0 with a_n=(-5)+(-3)n. List the prefix [a_0,…,a_(L-1)] of length L=8; use [] when L=0.
[-5, -8, -11, -14, -17, -20, -23, -26]
Take exactly 8 term(s) in index order beginning at 0. The explicit-formula prefix is [-5, -8, -11, -14, -17, -20, -23, -26], not a guessed continuation rule.
Reviewed step-by-step derivation
Sequence · qf-sequence-prefix · candidate
The real sequence is indexed from 0 with a_n=(-4)+(-2)n. List the prefix [a_0,…,a_(L-1)] of length L=1; use [] when L=0.
[-4]
Take exactly 1 term(s) in index order beginning at 0. The explicit-formula prefix is [-4], not a guessed continuation rule.
Reviewed step-by-step derivation
Sequence · qf-sequence-prefix · candidate
The real sequence is indexed from 0 with a_n=(-3)+(-1)n. List the prefix [a_0,…,a_(L-1)] of length L=2; use [] when L=0.
[-3, -4]
Take exactly 2 term(s) in index order beginning at 0. The explicit-formula prefix is [-3, -4], not a guessed continuation rule.
Reviewed step-by-step derivation
Sequence · qf-sequence-prefix · candidate
The real sequence is indexed from 0 with a_n=(-2)+(0)n. List the prefix [a_0,…,a_(L-1)] of length L=3; use [] when L=0.
[-2, -2, -2]
Take exactly 3 term(s) in index order beginning at 0. The explicit-formula prefix is [-2, -2, -2], not a guessed continuation rule.
Reviewed step-by-step derivation
Sequence · qf-sequence-prefix · candidate
The real sequence is indexed from 0 with a_n=(-1)+(1)n. List the prefix [a_0,…,a_(L-1)] of length L=4; use [] when L=0.
[-1, 0, 1, 2]
Take exactly 4 term(s) in index order beginning at 0. The explicit-formula prefix is [-1, 0, 1, 2], not a guessed continuation rule.
Reviewed step-by-step derivation
Sequence · qf-sequence-prefix · candidate
The real sequence is indexed from 0 with a_n=(0)+(2)n. List the prefix [a_0,…,a_(L-1)] of length L=5; use [] when L=0.
[0, 2, 4, 6, 8]
Take exactly 5 term(s) in index order beginning at 0. The explicit-formula prefix is [0, 2, 4, 6, 8], not a guessed continuation rule.
Reviewed step-by-step derivation
Sequence · qf-sequence-prefix · candidate
The real sequence is indexed from 0 with a_n=(1)+(3)n. List the prefix [a_0,…,a_(L-1)] of length L=6; use [] when L=0.
[1, 4, 7, 10, 13, 16]
Take exactly 6 term(s) in index order beginning at 0. The explicit-formula prefix is [1, 4, 7, 10, 13, 16], not a guessed continuation rule.
Reviewed step-by-step derivation
Sequence · qf-sequence-prefix · candidate
The real sequence is indexed from 0 with a_n=(2)+(-3)n. List the prefix [a_0,…,a_(L-1)] of length L=7; use [] when L=0.
[2, -1, -4, -7, -10, -13, -16]
Take exactly 7 term(s) in index order beginning at 0. The explicit-formula prefix is [2, -1, -4, -7, -10, -13, -16], not a guessed continuation rule.
Reviewed step-by-step derivation
Sequence · qf-sequence-prefix · candidate
The real sequence is indexed from 0 with a_n=(3)+(-2)n. List the prefix [a_0,…,a_(L-1)] of length L=8; use [] when L=0.
[3, 1, -1, -3, -5, -7, -9, -11]
Take exactly 8 term(s) in index order beginning at 0. The explicit-formula prefix is [3, 1, -1, -3, -5, -7, -9, -11], not a guessed continuation rule.
Reviewed step-by-step derivation
Sequence · qf-sequence-prefix · candidate
The real sequence is indexed from 0 with a_n=(4)+(-1)n. List the prefix [a_0,…,a_(L-1)] of length L=1; use [] when L=0.
[4]
Take exactly 1 term(s) in index order beginning at 0. The explicit-formula prefix is [4], not a guessed continuation rule.
Reviewed step-by-step derivation
Sequence · qf-sequence-prefix · candidate
The real sequence is indexed from 0 with a_n=(5)+(0)n. List the prefix [a_0,…,a_(L-1)] of length L=2; use [] when L=0.
[5, 5]
Take exactly 2 term(s) in index order beginning at 0. The explicit-formula prefix is [5, 5], not a guessed continuation rule.
Reviewed step-by-step derivation
Sequence · qf-sequence-prefix · candidate
The real sequence is indexed from 0 with a_n=(6)+(1)n. List the prefix [a_0,…,a_(L-1)] of length L=3; use [] when L=0.
[6, 7, 8]
Take exactly 3 term(s) in index order beginning at 0. The explicit-formula prefix is [6, 7, 8], not a guessed continuation rule.
Reviewed step-by-step derivation
Sequence · qf-sequence-prefix · candidate
The real sequence is indexed from 0 with a_n=(7)+(2)n. List the prefix [a_0,…,a_(L-1)] of length L=4; use [] when L=0.
[7, 9, 11, 13]
Take exactly 4 term(s) in index order beginning at 0. The explicit-formula prefix is [7, 9, 11, 13], not a guessed continuation rule.
Reviewed step-by-step derivation
Sequence · qf-sequence-prefix · candidate
The real sequence is indexed from 0 with a_n=(8)+(3)n. List the prefix [a_0,…,a_(L-1)] of length L=5; use [] when L=0.
[8, 11, 14, 17, 20]
Take exactly 5 term(s) in index order beginning at 0. The explicit-formula prefix is [8, 11, 14, 17, 20], not a guessed continuation rule.
Reviewed step-by-step derivation
Sequence · qf-sequence-prefix · candidate
The real sequence is indexed from 0 with a_n=(9)+(-3)n. List the prefix [a_0,…,a_(L-1)] of length L=6; use [] when L=0.
[9, 6, 3, 0, -3, -6]
Take exactly 6 term(s) in index order beginning at 0. The explicit-formula prefix is [9, 6, 3, 0, -3, -6], not a guessed continuation rule.
Reviewed step-by-step derivation
Sequence · qf-sequence-prefix · candidate
The real sequence is indexed from 0 with a_n=(10)+(-2)n. List the prefix [a_0,…,a_(L-1)] of length L=7; use [] when L=0.
[10, 8, 6, 4, 2, 0, -2]
Take exactly 7 term(s) in index order beginning at 0. The explicit-formula prefix is [10, 8, 6, 4, 2, 0, -2], not a guessed continuation rule.
Reviewed step-by-step derivation
Sequence · qf-sequence-prefix · candidate
The real sequence is indexed from 0 with a_n=(11)+(-1)n. List the prefix [a_0,…,a_(L-1)] of length L=8; use [] when L=0.
[11, 10, 9, 8, 7, 6, 5, 4]
Take exactly 8 term(s) in index order beginning at 0. The explicit-formula prefix is [11, 10, 9, 8, 7, 6, 5, 4], not a guessed continuation rule.
Reviewed step-by-step derivation
Sequence · qf-sequence-prefix · candidate
The real sequence is indexed from 0 with a_n=(12)+(0)n. List the prefix [a_0,…,a_(L-1)] of length L=1; use [] when L=0.
[12]
Take exactly 1 term(s) in index order beginning at 0. The explicit-formula prefix is [12], not a guessed continuation rule.
Reviewed step-by-step derivation
Sequence · qf-sequence-prefix · candidate
A finite real sequence has length L=0 and zero-based valid indices satisfying 0≤n<L. Is n=-1 a valid index?
No
Check 0≤-1<0; the answer is No. A sequence of length 0 has no valid indices.
Reviewed step-by-step derivation
Sequence · qf-sequence-finite-index-validity · candidate
A finite real sequence has length L=0 and zero-based valid indices satisfying 0≤n<L. Is n=0 a valid index?
No
Check 0≤0<0; the answer is No. A sequence of length 0 has no valid indices.
Reviewed step-by-step derivation
Sequence · qf-sequence-finite-index-validity · candidate
A finite real sequence has length L=0 and zero-based valid indices satisfying 0≤n<L. Is n=1 a valid index?
No
Check 0≤1<0; the answer is No. A sequence of length 0 has no valid indices.
Reviewed step-by-step derivation
Sequence · qf-sequence-finite-index-validity · candidate
A finite real sequence has length L=1 and zero-based valid indices satisfying 0≤n<L. Is n=-1 a valid index?
No
Check 0≤-1<1; the answer is No. A sequence of length 0 has no valid indices.
Reviewed step-by-step derivation
Sequence · qf-sequence-finite-index-validity · candidate
A finite real sequence has length L=1 and zero-based valid indices satisfying 0≤n<L. Is n=0 a valid index?
Yes
Check 0≤0<1; the answer is Yes. A sequence of length 0 has no valid indices.
Reviewed step-by-step derivation
Sequence · qf-sequence-finite-index-validity · candidate
A finite real sequence has length L=1 and zero-based valid indices satisfying 0≤n<L. Is n=1 a valid index?
No
Check 0≤1<1; the answer is No. A sequence of length 0 has no valid indices.
Reviewed step-by-step derivation
Sequence · qf-sequence-finite-index-validity · candidate
A finite real sequence has length L=1 and zero-based valid indices satisfying 0≤n<L. Is n=2 a valid index?
No
Check 0≤2<1; the answer is No. A sequence of length 0 has no valid indices.
Reviewed step-by-step derivation
Sequence · qf-sequence-finite-index-validity · candidate
A finite real sequence has length L=2 and zero-based valid indices satisfying 0≤n<L. Is n=0 a valid index?
Yes
Check 0≤0<2; the answer is Yes. A sequence of length 0 has no valid indices.
Reviewed step-by-step derivation
Sequence · qf-sequence-finite-index-validity · candidate
A finite real sequence has length L=2 and zero-based valid indices satisfying 0≤n<L. Is n=1 a valid index?
Yes
Check 0≤1<2; the answer is Yes. A sequence of length 0 has no valid indices.
Reviewed step-by-step derivation
Sequence · qf-sequence-finite-index-validity · candidate
A finite real sequence has length L=2 and zero-based valid indices satisfying 0≤n<L. Is n=2 a valid index?
No
Check 0≤2<2; the answer is No. A sequence of length 0 has no valid indices.
Reviewed step-by-step derivation
Sequence · qf-sequence-finite-index-validity · candidate
A finite real sequence has length L=2 and zero-based valid indices satisfying 0≤n<L. Is n=-2 a valid index?
No
Check 0≤-2<2; the answer is No. A sequence of length 0 has no valid indices.
Reviewed step-by-step derivation
Sequence · qf-sequence-finite-index-validity · candidate
A finite real sequence has length L=3 and zero-based valid indices satisfying 0≤n<L. Is n=2 a valid index?
Yes
Check 0≤2<3; the answer is Yes. A sequence of length 0 has no valid indices.
Reviewed step-by-step derivation
Sequence · qf-sequence-finite-index-validity · candidate
A finite real sequence has length L=3 and zero-based valid indices satisfying 0≤n<L. Is n=3 a valid index?
No
Check 0≤3<3; the answer is No. A sequence of length 0 has no valid indices.
Reviewed step-by-step derivation
Sequence · qf-sequence-finite-index-validity · candidate
A finite real sequence has length L=3 and zero-based valid indices satisfying 0≤n<L. Is n=4 a valid index?
No
Check 0≤4<3; the answer is No. A sequence of length 0 has no valid indices.
Reviewed step-by-step derivation
Sequence · qf-sequence-finite-index-validity · candidate
A finite real sequence has length L=3 and zero-based valid indices satisfying 0≤n<L. Is n=-1 a valid index?
No
Check 0≤-1<3; the answer is No. A sequence of length 0 has no valid indices.
Reviewed step-by-step derivation
Sequence · qf-sequence-finite-index-validity · candidate
A finite real sequence has length L=5 and zero-based valid indices satisfying 0≤n<L. Is n=0 a valid index?
Yes
Check 0≤0<5; the answer is Yes. A sequence of length 0 has no valid indices.
Reviewed step-by-step derivation
Sequence · qf-sequence-finite-index-validity · candidate
A finite real sequence has length L=5 and zero-based valid indices satisfying 0≤n<L. Is n=4 a valid index?
Yes
Check 0≤4<5; the answer is Yes. A sequence of length 0 has no valid indices.
Reviewed step-by-step derivation
Sequence · qf-sequence-finite-index-validity · candidate
A finite real sequence has length L=5 and zero-based valid indices satisfying 0≤n<L. Is n=5 a valid index?
No
Check 0≤5<5; the answer is No. A sequence of length 0 has no valid indices.
Reviewed step-by-step derivation
Sequence · qf-sequence-finite-index-validity · candidate
A finite real sequence has length L=5 and zero-based valid indices satisfying 0≤n<L. Is n=6 a valid index?
No
Check 0≤6<5; the answer is No. A sequence of length 0 has no valid indices.
Reviewed step-by-step derivation
Sequence · qf-sequence-finite-index-validity · candidate
A finite real sequence has length L=10 and zero-based valid indices satisfying 0≤n<L. Is n=-1 a valid index?
No
Check 0≤-1<10; the answer is No. A sequence of length 0 has no valid indices.
Reviewed step-by-step derivation
Sequence · qf-sequence-finite-index-validity · candidate
A finite real sequence has length L=10 and zero-based valid indices satisfying 0≤n<L. Is n=0 a valid index?
Yes
Check 0≤0<10; the answer is Yes. A sequence of length 0 has no valid indices.
Reviewed step-by-step derivation
Sequence · qf-sequence-finite-index-validity · candidate
A finite real sequence has length L=10 and zero-based valid indices satisfying 0≤n<L. Is n=9 a valid index?
Yes
Check 0≤9<10; the answer is Yes. A sequence of length 0 has no valid indices.
Reviewed step-by-step derivation
Sequence · qf-sequence-finite-index-validity · candidate
A finite real sequence has length L=10 and zero-based valid indices satisfying 0≤n<L. Is n=10 a valid index?
No
Check 0≤10<10; the answer is No. A sequence of length 0 has no valid indices.
Reviewed step-by-step derivation
Sequence · qf-sequence-finite-index-validity · candidate
A finite real sequence has length L=10 and zero-based valid indices satisfying 0≤n<L. Is n=11 a valid index?
No
Check 0≤11<10; the answer is No. A sequence of length 0 has no valid indices.
Reviewed step-by-step derivation
Sequence · qf-sequence-finite-index-validity · candidate
A finite real sequence has length L=100 and zero-based valid indices satisfying 0≤n<L. Is n=99 a valid index?
Yes
Check 0≤99<100; the answer is Yes. A sequence of length 0 has no valid indices.
Reviewed step-by-step derivation
Sequence · qf-sequence-finite-index-validity · candidate
Canonical question JSONL · English JSONL · Validation report