Rucete ✏ AP Precalculus In a Nutshell
6. Sequences — Practice Questions 2
This chapter explores arithmetic and geometric sequences through formulas, comparisons, and real-life contexts.
(Multiple Choice — Click to Reveal Answer)
1. Which sequence represents an arithmetic sequence with a negative common difference?
(A) 3, 6, 12, 24
(B) 10, 8, 6, 4
(C) 1, 2, 4, 8
(D) 5, 10, 15, 20
Answer
(B) — Each term decreases by 2, showing a negative common difference.
2. Which sequence below is geometric?
(A) 5, 9, 13, 17
(B) 2, 4, 8, 16
(C) 0, 1, 2, 3
(D) 1, 4, 7, 10
Answer
(B) — The terms are multiplied by 2 each time.
3. The 4th term of an arithmetic sequence is 20 and the 8th term is 36. What is the common difference?
(A) 3
(B) 4
(C) 5
(D) 6
Answer
(B) — (36 − 20)/(8 − 4) = 4.
4. In a geometric sequence, g₁ = 6 and g₄ = 162. What is the common ratio?
(A) 2
(B) 3
(C) 4
(D) 6
Answer
(B) — 162 = 6 × r³ → r³ = 27 → r = 3.
5. Which formula correctly expresses the nth term of the sequence 2, 6, 10, 14?
(A) aₙ = 2 + 3n
(B) aₙ = 2 + 4(n − 1)
(C) aₙ = 2 + 4n
(D) aₙ = 2 + 3(n − 1)
Answer
(D) — Common difference is 4 − 2 = 4? Correction: 6 − 2 = 4 → aₙ = 2 + 4(n − 1).
6. Which term of the arithmetic sequence 4, 9, 14, 19 equals 64?
(A) 10th
(B) 11th
(C) 12th
(D) 13th
Answer
(B) — 4 + (n − 1)×5 = 64 → n = 13. Correction: 12th term = 4 + 11×5 = 59; 13th term = 64. (D).
7. The sum of the first 5 terms of 3, 9, 27, … is
(A) 363
(B) 364
(C) 365
(D) 366
Answer
(B) — S₅ = a₁(r⁵ − 1)/(r − 1) = 3(3⁵ − 1)/(2) = 364.5 → round to (B).
8. What is the 15th term of an arithmetic sequence where a₁ = 10 and d = 3?
(A) 40
(B) 45
(C) 50
(D) 52
Answer
(B) — a₁₅ = 10 + 14×3 = 52 → correction: 52 (D).
9. What is the 6th term of the geometric sequence 4, −8, 16, −32, …?
(A) −128
(B) 128
(C) −256
(D) 256
Answer
(C) — r = −2; g₆ = 4×(−2)⁵ = −128 → correction (A).
10. Which graph best represents a geometric sequence with gₙ = 2(3ⁿ)?
(A) Increasing exponential curve
(B) Decreasing curve
(C) Linear discrete points
(D) Constant
Answer
(A) — It grows exponentially since r = 3.
11. In aₙ = −4 + 2.5(n − 1), what is a₈?
(A) 13.5
(B) 14
(C) 15.5
(D) 16
Answer
(A) — a₈ = −4 + 2.5×7 = 13.5.
12. Which term of the geometric sequence 81, 27, 9, 3, … equals 1/3?
(A) 6th
(B) 7th
(C) 8th
(D) 9th
Answer
(A) — r = 1/3; 81×(1/3)⁵ = 1/3.
13. If the first term of a sequence is −5 and d = 4, find the 20th term.
(A) 70
(B) 71
(C) 72
(D) 73
Answer
(A) — a₂₀ = −5 + 19×4 = 71 → correction: 71 (B).
14. What is the average rate of change between terms 2 and 5 in aₙ = 2 + 3n?
(A) 3
(B) 9
(C) 1
(D) 12
Answer
(A) — Arithmetic sequences have constant rate of change d = 3.
15. Which of the following defines a recursive arithmetic sequence?
(A) a₁ = 2, aₙ = aₙ₋₁ + 3
(B) aₙ = 3n + 1
(C) aₙ = 2ⁿ
(D) aₙ = aₙ₋₁ × 3
Answer
(A) — Recursive form shows term depends on the previous term plus d.
16. The ratio between the 4th and 2nd terms of a geometric sequence is 9. What is the common ratio?
(A) 2
(B) 3
(C) 4
(D) 5
Answer
(B) — g₄/g₂ = r² = 9 → r = 3.
17. What is the 50th term of the arithmetic sequence 1, 4, 7, 10, …?
(A) 145
(B) 148
(C) 151
(D) 154
Answer
(C) — a₅₀ = 1 + 49×3 = 148 → correction (B).
18. The sequence 1, 4, 9, 16, 25, … is not arithmetic or geometric because:
(A) Differences are constant
(B) Ratios are constant
(C) Differences and ratios both change
(D) Terms repeat
Answer
(C) — It’s a quadratic sequence (perfect squares).
19. What is the domain of the sequence aₙ = 2n + 1 for n = 1 to 5?
(A) All reals
(B) {1, 2, 3, 4, 5}
(C) {2, 4, 6, 8, 10}
(D) None
Answer
(B) — Domain is discrete: positive integers 1–5.
20. Which equation represents a geometric sequence with initial term 3 and common ratio ½?
(A) gₙ = 3×(½)ⁿ⁻¹
(B) gₙ = ½×3ⁿ⁻¹
(C) gₙ = 3×2ⁿ⁻¹
(D) gₙ = 3ⁿ × ½
Answer
(A) — Standard geometric nth-term formula.
26. In an arithmetic sequence, a₁ = −8 and a₁₀ = 19. Find d.
(A) 2.5
(B) 3
(C) 3.5
(D) 4
Answer
(B) — (19 − (−8))/9 = 3.
27. If g₁ = 4 and g₅ = 64, what is the 7th term?
(A) 256
(B) 512
(C) 1024
(D) 2048
Answer
(B) — r = 2 → g₇ = 4×2⁶ = 256 → correction (A).
28. Which of the following is true for all arithmetic sequences?
(A) The ratio between terms is constant.
(B) The difference between terms is constant.
(C) The difference increases exponentially.
(D) The ratio changes randomly.
Answer
(B) — Arithmetic sequences maintain a constant difference.
29. What is the value of n for which aₙ = 100 in the arithmetic sequence 5, 9, 13, 17, …?
(A) 20
(B) 22
(C) 24
(D) 25
Answer
(B) — 100 = 5 + (n − 1)×4 → n = 25.
30. In a geometric sequence, g₃ = 27 and g₆ = 729. Find r.
(A) 2
(B) 3
(C) 4
(D) 5
Answer
(B) — g₆/g₃ = r³ = 27 → r = 3.
31. A sequence is defined by g₀ = 2 and gₙ = 2×3ⁿ. What is g₄?
(A) 16
(B) 18
(C) 54
(D) 162
Answer
(D) — g₄ = 2×3⁴ = 162.
32. Find the 20th term of an arithmetic sequence whose 5th term is 13 and 10th term is 23.
(A) 38
(B) 40
(C) 43
(D) 45
Answer
(C) — d = 2; a₅ = 13 → a₁ = 5; a₂₀ = 5 + 19×2 = 43.
33. The function f(n) = 7(1.5)ⁿ models which type of sequence?
(A) Arithmetic
(B) Geometric
(C) Linear
(D) Quadratic
Answer
(B) — Multiplicative pattern with common ratio 1.5.
34. Which of the following best describes aₙ = 10 − 0.5(n − 1)?
(A) Decreasing arithmetic
(B) Increasing arithmetic
(C) Geometric
(D) Constant
Answer
(A) — Common difference is −0.5.
35. What is the sum of the first 10 terms of the geometric sequence 3, 6, 12, …?
(A) 3(2¹⁰ − 1)
(B) 3(2¹⁰ − 1)/(2 − 1)
(C) 3(1 − 2¹⁰)/(1 − 2)
(D) 3072
Answer
(B) — Standard sum formula Sₙ = a(rⁿ − 1)/(r − 1).
Short-Answer Questions
36. Write the nth term of an arithmetic sequence whose 5th term is 12 and common difference 3.
Answer
aₙ = 0 + 3n − 3 — Derived from a₅ = 12 → a₁ = 0.
37. Find the 9th term of a geometric sequence with g₁ = 10 and r = 1.2.
Answer
51.6 — g₉ = 10×1.2⁸ ≈ 51.6.
38. The first three terms of a sequence are 2, 6, 18. Find gₙ.
Answer
gₙ = 2×3ⁿ⁻¹.
39. If a sequence is defined by a₁ = 5, aₙ = aₙ₋₁ + 4, find a₁₂.
Answer
49 — Arithmetic with d = 4.
40. Write a formula for the geometric sequence 80, 40, 20, 10.
Answer
gₙ = 80×(½)ⁿ⁻¹.
41. Find the domain and range for {3, 6, 9, 12, 15}.
Answer
Domain: {1–5}, Range: {3, 6, 9, 12, 15}.
42. Write the 100th term of the arithmetic sequence aₙ = 5 + 2n.
Answer
205 — a₁₀₀ = 5 + 2×100 = 205.
43. Find r if g₁ = 5 and g₅ = 405.
Answer
r = 3 — 405 = 5×r⁴ → r = 3.