# Sequence Foundations 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. ## 1. Term of a Zero-Based Explicit Sequence 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. View answer and explanation-12 Substitute the explicit zero-based index: a_0=-12+(-3)×0=-12. No limit claim is involved. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r05-sequence-explicit-term-0001/) Sequence · qf-sequence-explicit-term · candidate ## 2. Term of a Zero-Based Explicit Sequence 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. View answer and explanation-13 Substitute the explicit zero-based index: a_1=-11+(-2)×1=-13. No limit claim is involved. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r05-sequence-explicit-term-0002/) Sequence · qf-sequence-explicit-term · candidate ## 3. Term of a Zero-Based Explicit Sequence 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. View answer and explanation-12 Substitute the explicit zero-based index: a_2=-10+(-1)×2=-12. No limit claim is involved. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r05-sequence-explicit-term-0003/) Sequence · qf-sequence-explicit-term · candidate ## 4. Term of a Zero-Based Explicit Sequence 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. View answer and explanation-9 Substitute the explicit zero-based index: a_3=-9+(0)×3=-9. No limit claim is involved. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r05-sequence-explicit-term-0004/) Sequence · qf-sequence-explicit-term · candidate ## 5. Term of a Zero-Based Explicit Sequence 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. View answer and explanation-4 Substitute the explicit zero-based index: a_4=-8+(1)×4=-4. No limit claim is involved. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r05-sequence-explicit-term-0005/) Sequence · qf-sequence-explicit-term · candidate ## 6. Term of a Zero-Based Explicit Sequence 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. View answer and explanation3 Substitute the explicit zero-based index: a_5=-7+(2)×5=3. No limit claim is involved. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r05-sequence-explicit-term-0006/) Sequence · qf-sequence-explicit-term · candidate ## 7. Term of a Zero-Based Explicit Sequence 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. View answer and explanation12 Substitute the explicit zero-based index: a_6=-6+(3)×6=12. No limit claim is involved. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r05-sequence-explicit-term-0007/) Sequence · qf-sequence-explicit-term · candidate ## 8. Term of a Zero-Based Explicit Sequence 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. View answer and explanation-26 Substitute the explicit zero-based index: a_7=-5+(-3)×7=-26. No limit claim is involved. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r05-sequence-explicit-term-0008/) Sequence · qf-sequence-explicit-term · candidate ## 9. Term of a Zero-Based Explicit Sequence 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. View answer and explanation-20 Substitute the explicit zero-based index: a_8=-4+(-2)×8=-20. No limit claim is involved. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r05-sequence-explicit-term-0009/) Sequence · qf-sequence-explicit-term · candidate ## 10. Term of a Zero-Based Explicit Sequence 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. View answer and explanation-12 Substitute the explicit zero-based index: a_9=-3+(-1)×9=-12. No limit claim is involved. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r05-sequence-explicit-term-0010/) Sequence · qf-sequence-explicit-term · candidate ## 11. Term of a Zero-Based Explicit Sequence 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. View answer and explanation-2 Substitute the explicit zero-based index: a_0=-2+(0)×0=-2. No limit claim is involved. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r05-sequence-explicit-term-0011/) Sequence · qf-sequence-explicit-term · candidate ## 12. Term of a Zero-Based Explicit Sequence 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. View answer and explanation0 Substitute the explicit zero-based index: a_1=-1+(1)×1=0. No limit claim is involved. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r05-sequence-explicit-term-0012/) Sequence · qf-sequence-explicit-term · candidate ## 13. Term of a Zero-Based Explicit Sequence 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. View answer and explanation4 Substitute the explicit zero-based index: a_2=0+(2)×2=4. No limit claim is involved. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r05-sequence-explicit-term-0013/) Sequence · qf-sequence-explicit-term · candidate ## 14. Term of a Zero-Based Explicit Sequence 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. View answer and explanation10 Substitute the explicit zero-based index: a_3=1+(3)×3=10. No limit claim is involved. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r05-sequence-explicit-term-0014/) Sequence · qf-sequence-explicit-term · candidate ## 15. Term of a Zero-Based Explicit Sequence 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. View answer and explanation-10 Substitute the explicit zero-based index: a_4=2+(-3)×4=-10. No limit claim is involved. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r05-sequence-explicit-term-0015/) Sequence · qf-sequence-explicit-term · candidate ## 16. Term of a Zero-Based Explicit Sequence 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. View answer and explanation-7 Substitute the explicit zero-based index: a_5=3+(-2)×5=-7. No limit claim is involved. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r05-sequence-explicit-term-0016/) Sequence · qf-sequence-explicit-term · candidate ## 17. Term of a Zero-Based Explicit Sequence 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. View answer and explanation-2 Substitute the explicit zero-based index: a_6=4+(-1)×6=-2. No limit claim is involved. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r05-sequence-explicit-term-0017/) Sequence · qf-sequence-explicit-term · candidate ## 18. Term of a Zero-Based Explicit Sequence 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. View answer and explanation5 Substitute the explicit zero-based index: a_7=5+(0)×7=5. No limit claim is involved. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r05-sequence-explicit-term-0018/) Sequence · qf-sequence-explicit-term · candidate ## 19. Term of a Zero-Based Explicit Sequence 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. View answer and explanation14 Substitute the explicit zero-based index: a_8=6+(1)×8=14. No limit claim is involved. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r05-sequence-explicit-term-0019/) Sequence · qf-sequence-explicit-term · candidate ## 20. Term of a Zero-Based Explicit Sequence 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. View answer and explanation25 Substitute the explicit zero-based index: a_9=7+(2)×9=25. No limit claim is involved. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r05-sequence-explicit-term-0020/) Sequence · qf-sequence-explicit-term · candidate ## 21. Term of a Zero-Based Explicit Sequence 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. View answer and explanation8 Substitute the explicit zero-based index: a_0=8+(3)×0=8. No limit claim is involved. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r05-sequence-explicit-term-0021/) Sequence · qf-sequence-explicit-term · candidate ## 22. Term of a Zero-Based Explicit Sequence 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. View answer and explanation6 Substitute the explicit zero-based index: a_1=9+(-3)×1=6. No limit claim is involved. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r05-sequence-explicit-term-0022/) Sequence · qf-sequence-explicit-term · candidate ## 23. Term of a Zero-Based Explicit Sequence 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. View answer and explanation6 Substitute the explicit zero-based index: a_2=10+(-2)×2=6. No limit claim is involved. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r05-sequence-explicit-term-0023/) Sequence · qf-sequence-explicit-term · candidate ## 24. Term of a Zero-Based Explicit Sequence 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. View answer and explanation8 Substitute the explicit zero-based index: a_3=11+(-1)×3=8. No limit claim is involved. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r05-sequence-explicit-term-0024/) Sequence · qf-sequence-explicit-term · candidate ## 25. Term of a Zero-Based Explicit Sequence 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. View answer and explanation12 Substitute the explicit zero-based index: a_4=12+(0)×4=12. No limit claim is involved. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r05-sequence-explicit-term-0025/) Sequence · qf-sequence-explicit-term · candidate ## 26. Term of a Recurrence with an Initial Value 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. View answer and explanation12 Start from a_0 and perform exactly 0 recurrence update(s), obtaining a_0=12. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r05-sequence-recursive-term-0001/) Sequence · qf-sequence-recursive-term · candidate ## 27. Term of a Recurrence with an Initial Value 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. View answer and explanation5 Start from a_0 and perform exactly 3 recurrence update(s), obtaining a_3=5. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r05-sequence-recursive-term-0002/) Sequence · qf-sequence-recursive-term · candidate ## 28. Term of a Recurrence with an Initial Value 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. View answer and explanation4 Start from a_0 and perform exactly 6 recurrence update(s), obtaining a_6=4. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r05-sequence-recursive-term-0003/) Sequence · qf-sequence-recursive-term · candidate ## 29. Term of a Recurrence with an Initial Value 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. View answer and explanation9 Start from a_0 and perform exactly 9 recurrence update(s), obtaining a_9=9. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r05-sequence-recursive-term-0004/) Sequence · qf-sequence-recursive-term · candidate ## 30. Term of a Recurrence with an Initial Value 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. View answer and explanation20 Start from a_0 and perform exactly 12 recurrence update(s), obtaining a_12=20. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r05-sequence-recursive-term-0005/) Sequence · qf-sequence-recursive-term · candidate ## 31. Term of a Recurrence with an Initial Value 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. View answer and explanation11 Start from a_0 and perform exactly 2 recurrence update(s), obtaining a_2=11. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r05-sequence-recursive-term-0006/) Sequence · qf-sequence-recursive-term · candidate ## 32. Term of a Recurrence with an Initial Value 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. View answer and explanation21 Start from a_0 and perform exactly 5 recurrence update(s), obtaining a_5=21. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r05-sequence-recursive-term-0007/) Sequence · qf-sequence-recursive-term · candidate ## 33. Term of a Recurrence with an Initial Value 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. View answer and explanation-19 Start from a_0 and perform exactly 8 recurrence update(s), obtaining a_8=-19. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r05-sequence-recursive-term-0008/) Sequence · qf-sequence-recursive-term · candidate ## 34. Term of a Recurrence with an Initial Value 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. View answer and explanation-18 Start from a_0 and perform exactly 11 recurrence update(s), obtaining a_11=-18. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r05-sequence-recursive-term-0009/) Sequence · qf-sequence-recursive-term · candidate ## 35. Term of a Recurrence with an Initial Value 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. View answer and explanation2 Start from a_0 and perform exactly 1 recurrence update(s), obtaining a_1=2. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r05-sequence-recursive-term-0010/) Sequence · qf-sequence-recursive-term · candidate ## 36. Term of a Recurrence with an Initial Value 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. View answer and explanation2 Start from a_0 and perform exactly 4 recurrence update(s), obtaining a_4=2. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r05-sequence-recursive-term-0011/) Sequence · qf-sequence-recursive-term · candidate ## 37. Term of a Recurrence with an Initial Value 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. View answer and explanation8 Start from a_0 and perform exactly 7 recurrence update(s), obtaining a_7=8. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r05-sequence-recursive-term-0012/) Sequence · qf-sequence-recursive-term · candidate ## 38. Term of a Recurrence with an Initial Value 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. View answer and explanation20 Start from a_0 and perform exactly 10 recurrence update(s), obtaining a_10=20. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r05-sequence-recursive-term-0013/) Sequence · qf-sequence-recursive-term · candidate ## 39. Term of a Recurrence with an Initial Value 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. View answer and explanation-1 Start from a_0 and perform exactly 0 recurrence update(s), obtaining a_0=-1. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r05-sequence-recursive-term-0014/) Sequence · qf-sequence-recursive-term · candidate ## 40. Term of a Recurrence with an Initial Value 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. View answer and explanation-11 Start from a_0 and perform exactly 3 recurrence update(s), obtaining a_3=-11. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r05-sequence-recursive-term-0015/) Sequence · qf-sequence-recursive-term · candidate ## 41. Term of a Recurrence with an Initial Value 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. View answer and explanation-15 Start from a_0 and perform exactly 6 recurrence update(s), obtaining a_6=-15. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r05-sequence-recursive-term-0016/) Sequence · qf-sequence-recursive-term · candidate ## 42. Term of a Recurrence with an Initial Value 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. View answer and explanation-13 Start from a_0 and perform exactly 9 recurrence update(s), obtaining a_9=-13. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r05-sequence-recursive-term-0017/) Sequence · qf-sequence-recursive-term · candidate ## 43. Term of a Recurrence with an Initial Value 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. View answer and explanation-5 Start from a_0 and perform exactly 12 recurrence update(s), obtaining a_12=-5. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r05-sequence-recursive-term-0018/) Sequence · qf-sequence-recursive-term · candidate ## 44. Term of a Recurrence with an Initial Value 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. View answer and explanation-4 Start from a_0 and perform exactly 2 recurrence update(s), obtaining a_2=-4. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r05-sequence-recursive-term-0019/) Sequence · qf-sequence-recursive-term · candidate ## 45. Term of a Recurrence with an Initial Value 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. View answer and explanation3 Start from a_0 and perform exactly 5 recurrence update(s), obtaining a_5=3. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r05-sequence-recursive-term-0020/) Sequence · qf-sequence-recursive-term · candidate ## 46. Term of a Recurrence with an Initial Value 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. View answer and explanation16 Start from a_0 and perform exactly 8 recurrence update(s), obtaining a_8=16. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r05-sequence-recursive-term-0021/) Sequence · qf-sequence-recursive-term · candidate ## 47. Term of a Recurrence with an Initial Value 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. View answer and explanation-42 Start from a_0 and perform exactly 11 recurrence update(s), obtaining a_11=-42. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r05-sequence-recursive-term-0022/) Sequence · qf-sequence-recursive-term · candidate ## 48. Term of a Recurrence with an Initial Value 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. View answer and explanation-12 Start from a_0 and perform exactly 1 recurrence update(s), obtaining a_1=-12. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r05-sequence-recursive-term-0023/) Sequence · qf-sequence-recursive-term · candidate ## 49. Term of a Recurrence with an Initial Value 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. View answer and explanation-15 Start from a_0 and perform exactly 4 recurrence update(s), obtaining a_4=-15. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r05-sequence-recursive-term-0024/) Sequence · qf-sequence-recursive-term · candidate ## 50. Term of a Recurrence with an Initial Value 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. View answer and explanation-12 Start from a_0 and perform exactly 7 recurrence update(s), obtaining a_7=-12. [Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260930-r05-sequence-recursive-term-0025/) Sequence · qf-sequence-recursive-term · candidate ## 51. Finite Prefix of an Explicit Sequence 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. View answer and explanation[] 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](https://ocme.evemisslab.com/en/questions/q-20260930-r05-sequence-prefix-0001/) Sequence · qf-sequence-prefix · candidate ## 52. Finite Prefix of an Explicit Sequence 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. View answer and explanation[-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](https://ocme.evemisslab.com/en/questions/q-20260930-r05-sequence-prefix-0002/) Sequence · qf-sequence-prefix · candidate ## 53. Finite Prefix of an Explicit Sequence 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. View answer and explanation[-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](https://ocme.evemisslab.com/en/questions/q-20260930-r05-sequence-prefix-0003/) Sequence · qf-sequence-prefix · candidate ## 54. Finite Prefix of an Explicit Sequence 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. View answer and explanation[-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](https://ocme.evemisslab.com/en/questions/q-20260930-r05-sequence-prefix-0004/) Sequence · qf-sequence-prefix · candidate ## 55. Finite Prefix of an Explicit Sequence 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. View answer and explanation[-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](https://ocme.evemisslab.com/en/questions/q-20260930-r05-sequence-prefix-0005/) Sequence · qf-sequence-prefix · candidate ## 56. Finite Prefix of an Explicit Sequence 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. View answer and explanation[-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](https://ocme.evemisslab.com/en/questions/q-20260930-r05-sequence-prefix-0006/) Sequence · qf-sequence-prefix · candidate ## 57. Finite Prefix of an Explicit Sequence 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. View answer and explanation[-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](https://ocme.evemisslab.com/en/questions/q-20260930-r05-sequence-prefix-0007/) Sequence · qf-sequence-prefix · candidate ## 58. Finite Prefix of an Explicit Sequence 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. View answer and explanation[-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](https://ocme.evemisslab.com/en/questions/q-20260930-r05-sequence-prefix-0008/) Sequence · qf-sequence-prefix · candidate ## 59. Finite Prefix of an Explicit Sequence 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. View answer and explanation[-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](https://ocme.evemisslab.com/en/questions/q-20260930-r05-sequence-prefix-0009/) Sequence · qf-sequence-prefix · candidate ## 60. Finite Prefix of an Explicit Sequence 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. View answer and explanation[-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](https://ocme.evemisslab.com/en/questions/q-20260930-r05-sequence-prefix-0010/) Sequence · qf-sequence-prefix · candidate ## 61. Finite Prefix of an Explicit Sequence 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. View answer and explanation[-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](https://ocme.evemisslab.com/en/questions/q-20260930-r05-sequence-prefix-0011/) Sequence · qf-sequence-prefix · candidate ## 62. Finite Prefix of an Explicit Sequence 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. View answer and explanation[-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](https://ocme.evemisslab.com/en/questions/q-20260930-r05-sequence-prefix-0012/) Sequence · qf-sequence-prefix · candidate ## 63. Finite Prefix of an Explicit Sequence 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. View answer and explanation[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](https://ocme.evemisslab.com/en/questions/q-20260930-r05-sequence-prefix-0013/) Sequence · qf-sequence-prefix · candidate ## 64. Finite Prefix of an Explicit Sequence 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. View answer and explanation[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](https://ocme.evemisslab.com/en/questions/q-20260930-r05-sequence-prefix-0014/) Sequence · qf-sequence-prefix · candidate ## 65. Finite Prefix of an Explicit Sequence 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. View answer and explanation[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](https://ocme.evemisslab.com/en/questions/q-20260930-r05-sequence-prefix-0015/) Sequence · qf-sequence-prefix · candidate ## 66. Finite Prefix of an Explicit Sequence 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. View answer and explanation[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](https://ocme.evemisslab.com/en/questions/q-20260930-r05-sequence-prefix-0016/) Sequence · qf-sequence-prefix · candidate ## 67. Finite Prefix of an Explicit Sequence 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. View answer and explanation[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](https://ocme.evemisslab.com/en/questions/q-20260930-r05-sequence-prefix-0017/) Sequence · qf-sequence-prefix · candidate ## 68. Finite Prefix of an Explicit Sequence 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. View answer and explanation[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](https://ocme.evemisslab.com/en/questions/q-20260930-r05-sequence-prefix-0018/) Sequence · qf-sequence-prefix · candidate ## 69. Finite Prefix of an Explicit Sequence 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. View answer and explanation[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](https://ocme.evemisslab.com/en/questions/q-20260930-r05-sequence-prefix-0019/) Sequence · qf-sequence-prefix · candidate ## 70. Finite Prefix of an Explicit Sequence 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. View answer and explanation[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](https://ocme.evemisslab.com/en/questions/q-20260930-r05-sequence-prefix-0020/) Sequence · qf-sequence-prefix · candidate ## 71. Finite Prefix of an Explicit Sequence 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. View answer and explanation[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](https://ocme.evemisslab.com/en/questions/q-20260930-r05-sequence-prefix-0021/) Sequence · qf-sequence-prefix · candidate ## 72. Finite Prefix of an Explicit Sequence 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. View answer and explanation[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](https://ocme.evemisslab.com/en/questions/q-20260930-r05-sequence-prefix-0022/) Sequence · qf-sequence-prefix · candidate ## 73. Finite Prefix of an Explicit Sequence 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. View answer and explanation[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](https://ocme.evemisslab.com/en/questions/q-20260930-r05-sequence-prefix-0023/) Sequence · qf-sequence-prefix · candidate ## 74. Finite Prefix of an Explicit Sequence 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. View answer and explanation[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](https://ocme.evemisslab.com/en/questions/q-20260930-r05-sequence-prefix-0024/) Sequence · qf-sequence-prefix · candidate ## 75. Finite Prefix of an Explicit Sequence 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. View answer and explanation[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](https://ocme.evemisslab.com/en/questions/q-20260930-r05-sequence-prefix-0025/) Sequence · qf-sequence-prefix · candidate ## 76. Index Boundary of a Finite Sequence A finite real sequence has length L=0 and zero-based valid indices satisfying 0≤n