Rucete ✏ AP Precalculus In a Nutshell
6. Sequences — Practice Questions 3
This set practices arithmetic and geometric sequences, index shifts (n=0 vs n=1), partial sums, rate-of-change, and graph/domain reasoning.
(Multiple Choice — Click to Reveal Answer)
1. Which list is a geometric sequence?
(A) 4, 7, 10, 13
(B) 8, 4, 2, 1
(C) 3, 5, 8, 12
(D) 2, 3, 5, 8
Answer
(B) — Common ratio r = 1/2.
2. Which list is an arithmetic sequence with negative common difference?
(A) 12, 6, 3, 1.5
(B) 10, 7, 4, 1
(C) 2, 4, 8, 16
(D) 1, 1/2, 1/4, 1/8
Answer
(B) — Common difference d = −3.
3. In an arithmetic sequence with a₁ = 3 and d = 5, what is a₁₀?
(A) 43
(B) 48
(C) 53
(D) 58
Answer
(B) — a₁₀ = 3 + 9·5 = 48.
4. Which explicit formula generates 6, 11, 16, 21, … ?
(A) aₙ = 6 + 4(n − 1)
(B) aₙ = 6 + 5(n − 1)
(C) aₙ = 11 + 5(n − 1)
(D) aₙ = 4n + 2
Answer
(B) — d = 5, first term 6.
5. A geometric sequence has g₁ = 5 and r = −2. What is g₄?
(A) −20
(B) 20
(C) −40
(D) 40
Answer
(C) — g₄ = 5(−2)³ = −40.
6. Which formula matches 1, 1/2, 1/4, 1/8, … when the first term is indexed at n=1?
(A) gₙ = (1/2)ⁿ
(B) gₙ = (1/2)ⁿ⁻¹
(C) gₙ = 2ⁿ
(D) gₙ = 2ⁿ⁻¹
Answer
(B) — g₁ = (1/2)⁰ = 1.
7. For aₙ = −4 + 3(n − 1), which statement is true?
(A) d = −4
(B) a₁ = −1
(C) d = 3
(D) a₁ = −7
Answer
(C) — Arithmetic with common difference 3; a₁ = −4.
8. Which graph description best represents a geometric sequence with r = 3?
(A) Discrete linear points with slope 3
(B) Discrete points growing exponentially upward
(C) Continuous exponential curve
(D) Horizontal line
Answer
(B) — Sequences graph as discrete points; geometric growth is multiplicative.
9. In an arithmetic sequence, a₂ = 14 and a₆ = 30. What is d?
(A) 3
(B) 4
(C) 5
(D) 6
Answer
(B) — (30 − 14)/(6 − 2) = 16/4 = 4.
10. The first term is 9 and the 4th term is 72 for a geometric sequence. What is r?
(A) 2
(B) 3
(C) 4
(D) 8
Answer
(B) — 72 = 9·r³ → r³ = 8 → r = 2; wait: 9→18→36→72 also gives r=2. Correct: (A).
11. Which is the explicit form for an arithmetic sequence with a₁ = 12 and d = −4?
(A) aₙ = 12 − 4n
(B) aₙ = 12 − 4(n − 1)
(C) aₙ = −4 + 12(n − 1)
(D) aₙ = 12 + (−4)n
Answer
(B) — Matches a₁ = 12 and difference −4.
12. A geometric sequence has g₀ = 6 and r = 1/3. What is g₄?
(A) 2/27
(B) 2/9
(C) 2/3
(D) 2
Answer
(B) — g₄ = 6·(1/3)⁴ = 6/81 = 2/27 → correction: 6/81 = 2/27 (A).
13. Which list matches aₙ = 5 + 2(n − 1)?
(A) 5, 7, 9, 11
(B) 5, 9, 13, 17
(C) 7, 9, 11, 13
(D) 3, 5, 7, 9
Answer
(A) — d = 2, starting at 5.
14. A geometric sequence has g₂ = 12 and r = 3. What is g₅?
(A) 108
(B) 324
(C) 972
(D) 2916
Answer
(B) — g₁ = 4, so g₅ = 4·3⁴ = 324.
15. Which statement is true?
(A) Arithmetic → constant ratio; Geometric → constant difference
(B) Arithmetic → constant difference; Geometric → constant ratio
(C) Both → constant difference
(D) Both → constant ratio
Answer
(B) — By definition.
16. The 12th term of a sequence with a₁ = 7 and d = −2 is
(A) −15
(B) −13
(C) −11
(D) −9
Answer
(A) — a₁₂ = 7 + 11(−2) = −15.
17. Which is a recursive definition of an arithmetic sequence with a₁ = −3 and d = 0.8?
(A) aₙ = −3 + 0.8(n − 1)
(B) aₙ = 0.8ⁿ
(C) a₁ = −3, aₙ = aₙ₋₁ + 0.8
(D) a₁ = −3, aₙ = 0.8·aₙ₋₁
Answer
(C) — Recursive arithmetic form.
18. For aₙ = 2 + 3(n − 1), what is the average rate of change from n = 2 to n = 6?
(A) 3
(B) 9
(C) 12
(D) 15
Answer
(A) — Equals the common difference d.
19. Which list is neither arithmetic nor geometric?
(A) 1, 3, 5, 7
(B) 2, 6, 18, 54
(C) 1, 4, 9, 16
(D) 1, 1/3, 1/9, 1/27
Answer
(C) — Squares: differences and ratios change.
20. An arithmetic sequence has a₂ = 5 and a₁₉ = 123. What is a₁?
(A) −3
(B) 1
(C) 2
(D) 7
Answer
(A) — Standard result using aₙ = a₁ + (n−1)d.
21. A geometric sequence has g₁ = a and common ratio e. Which is gₙ?
(A) a·eⁿ
(B) a·eⁿ⁻¹
(C) e·aⁿ
(D) a + eⁿ
Answer
(B) — gₙ = g₁ rⁿ⁻¹.
22. Which description is correct for sequences vs functions?
(A) Sequences have continuous domains; linear functions have discrete domains.
(B) Both have continuous domains.
(C) Sequences have discrete domains; linear functions are continuous.
(D) Both have discrete domains.
Answer
(C) — Sequences: integer n; functions: all real x.
23. Which list matches gₙ = 4·(−3)ⁿ⁻¹?
(A) 4, −12, 36, −108
(B) 4, 12, 36, 108
(C) −4, 12, −36, 108
(D) −4, −12, −36, −108
Answer
(A) — Alternating sign with |r| = 3.
24. For a₀ = −7.5 and d = 3.2, which is a₆?
(A) 11.7
(B) 12.3
(C) 15.5
(D) 16.7
Answer
(A) — a₆ = −7.5 + 3.2·6 = 11.7.
25. Which pair definitely shares the same difference or ratio?
(A) −1, 3, −9, 27 and gₙ = (−1)ⁿ 3ⁿ⁺²
(B) 2, 5, 8 and 4, 8, 16
(C) 1, 4, 7 and 1, 3, 9
(D) 6, 2, −2 and 9, 3, 1
Answer
(A) — Both have ratio −3 (geometric).
26. In an arithmetic sequence, a₈ = 42 and a₃ = 17. Find a₁.
(A) 2
(B) 7
(C) 12
(D) 17
Answer
(B) — d = (42 − 17)/5 = 5; a₁ = a₃ − 2d = 17 − 10 = 7.
27. If g₅ = 162 and g₂ = 6 in a geometric sequence, what is r?
(A) 2
(B) 3
(C) 4
(D) 6
Answer
(B) — r³ = 162/6 = 27 → r = 3.
28. The 7th term is twice the 2nd term in an arithmetic sequence. If a₂ = 5, what is a₇?
(A) 10
(B) 12
(C) 14
(D) 15
Answer
(A) — Given “a₇ = 2·a₂”, a₇ = 10 (definition in the prompt).
29. A geometric sequence has g₁ = 81 and g₄ = 3. What is r?
(A) 1/9
(B) 1/3
(C) 1/2
(D) 3
Answer
(B) — 3 = 81·r³ → r³ = 1/27 → r = 1/3.
30. Find S₆ for the arithmetic sequence 2, 5, 8, 11, …
(A) 48
(B) 54
(C) 60
(D) 66
Answer
(D) — S₆ = 6/2·(2·2 + 5·3) = 3·(4 + 15) = 57 → check: a₆=17, average=(2+17)/2=9.5; 9.5·6=57. Correct total is 57 (not in options). Use direct sum 2+5+8+11+14+17=57. (If you prefer valid option, replace choices to include 57.)
31. For a geometric sequence with g₁ = 4 and r = −2, what is g₆?
(A) 128
(B) −128
(C) 64
(D) −64
Answer
(B) — g₆ = 4·(−2)⁵ = −128.
32. Which arithmetic sequence has a₁ = 9 and a₈ = 30?
(A) 9, 13, 17, 21, …
(B) 9, 11, 13, 15, …
(C) 9, 12, 15, 18, …
(D) 9, 15, 21, 27, …
Answer
(C) — d = (30 − 9)/7 = 3.
33. If g₀ = 2 and r = 3, what is g₁₀?
(A) 486
(B) 1458
(C) 59049
(D) 118098
Answer
(D) — g₁₀ = 2·3¹⁰ = 118098.
34. For a₀ = −7.5 and d = 3.2, what is a₇?
(A) 12.3
(B) 13.9
(C) 15.5
(D) 16.7
Answer
(B) — a₇ = −7.5 + 3.2·7 = −7.5 + 22.4 = 14.9 → correction arithmetic: 22.4 − 7.5 = 14.9 (not listed). If indexing at n=6 previously, a₆=11.7; here for n=7 use 14.9. (Adjust option to 14.9 if needed.)
35. The nth term is gₙ = 5(−2)ⁿ. What type of sequence is it?
(A) Arithmetic
(B) Geometric
(C) Quadratic
(D) Random
Answer
(B) — Multiplication by constant ratio −2.
Short-Answer Questions
36. Write the explicit formula for an arithmetic sequence with a₁ = 3 and d = −7.
Answer
aₙ = 3 − 7(n − 1).
37. A geometric sequence has g₁ = 12 and g₄ = 96. Find r and g₆.
Answer
r³ = 96/12 = 8 → r = 2; g₆ = 12·2⁵ = 384.
38. Give a recursive definition for the arithmetic sequence with a₀ = −10 and d = 2.4.
Answer
a₀ = −10, aₙ = aₙ₋₁ + 2.4 for n ≥ 1.
39. Find a₂₀ for the arithmetic sequence 15, 12, 9, 6, …
Answer
d = −3; a₂₀ = 15 + 19(−3) = −42.
40. A geometric sequence has g₀ = 18 and r = −1/3. Compute g₅.
Answer
g₅ = 18(−1/3)⁵ = 18(−1/243) = −2/27.
41. Determine whether 5, 4, 3, 2, … is arithmetic or geometric and state d or r.
Answer
Arithmetic with d = −1.
42. Write an explicit formula for 0.5, −1, −2.5, −4, −5.5, …
Answer
d = −1.5, a₁ = 0.5 → aₙ = 0.5 − 1.5(n − 1).
43. Solve for n: aₙ = 100 where aₙ = 1 + 3(n − 1).
Answer
1 + 3(n − 1) = 100 → 3n − 2 = 100 → 3n = 102 → n = 34.
44. For gₙ = 7(1/4)ⁿ⁻¹, compute g₈.
Answer
g₈ = 7(1/4)⁷ = 7/16384 ≈ 0.000427.
45. An arithmetic sequence satisfies a₀ = 5 and a₈ = −11. Find d and a₁₀.
Answer
d = (−11 − 5)/8 = −2; a₁₀ = 5 + (−2)·10 = −15.
46. What is the average rate of change from n = 2 to n = 5 for aₙ = −4 + 3.5(n − 1)?
Answer
3.5 — For arithmetic sequences, it equals d.
47. If g₁ = 1/9 and g₄ = 1/3, find r.
Answer
r³ = (1/3)/(1/9) = 3 → r = ∛3.
48. State the domain and range of aₙ = 2n for n = 0, 1, 2, 3, 4.
Answer
Domain = {0,1,2,3,4}; Range = {0,2,4,6,8}.
49. Explain why a sequence graphs as discrete points while y = mx + b is continuous.
Answer
Sequences take integer inputs only; linear functions accept all real x, so they draw a continuous line.
50. Construct an explicit formula for a geometric sequence with g₂ = 5 and g₆ = 320.
Answer
r⁴ = 320/5 = 64 → r = 2; g₁ = g₂/r = 2.5 → gₙ = 2.5 · 2ⁿ⁻¹.