Rucete ✏ AP Precalculus In a Nutshell
6. Sequences — Practice Questions
This chapter introduces arithmetic and geometric sequences, their formulas, and their relationships to linear and exponential functions.
(Multiple Choice — Click to Reveal Answer)
1. Which of the following describes an arithmetic sequence?
(A) Each term is multiplied by a constant.
(B) Each term is divided by a constant.
(C) Each term is increased by a constant.
(D) Each term is squared.
Answer
(C) — Arithmetic sequences have a constant difference between consecutive terms.
2. What is the common difference of the sequence 4, 9, 14, 19, 24?
(A) 4
(B) 5
(C) 6
(D) 7
Answer
(B) — Each term increases by 5.
3. What is the 10th term of the arithmetic sequence with a₁ = 3 and d = 4?
(A) 39
(B) 40
(C) 43
(D) 44
Answer
(A) — a₁₀ = 3 + 9×4 = 39.
4. Which formula represents the nth term of an arithmetic sequence with first term 7 and common difference −2?
(A) aₙ = 7 − 2n
(B) aₙ = 7 − 2(n − 1)
(C) aₙ = −2n + 5
(D) aₙ = 2n + 7
Answer
(B) — Subtracting the difference (−2) after the first term gives aₙ = 7 − 2(n − 1).
5. What type of sequence is 3, −6, 12, −24, 48?
(A) Arithmetic
(B) Geometric
(C) Quadratic
(D) Recursive
Answer
(B) — The terms are multiplied by −2 each time.
6. What is the common ratio of the geometric sequence 2, 8, 32, 128?
(A) 2
(B) 3
(C) 4
(D) 6
Answer
(C) — Each term is multiplied by 4.
7. What is the 5th term of the geometric sequence with g₁ = 2 and r = 3?
(A) 18
(B) 24
(C) 54
(D) 162
Answer
(D) — g₅ = 2 × 3⁴ = 162.
8. Which expression gives the nth term of the geometric sequence 1, 4, 16, 64, …?
(A) gₙ = 1 + 4n
(B) gₙ = 4n
(C) gₙ = 4ⁿ
(D) gₙ = 1 × 4ⁿ⁻¹
Answer
(D) — For g₁ = 1 and r = 4, gₙ = 1 × 4ⁿ⁻¹.
9. In an arithmetic sequence, a₃ = 14 and a₆ = 23. What is the common difference?
(A) 2
(B) 3
(C) 4
(D) 5
Answer
(B) — (23 − 14) ÷ (6 − 3) = 3.
10. What is the first term of an arithmetic sequence whose 2nd term is 5 and common difference is 3?
(A) 2
(B) 3
(C) 4
(D) 5
Answer
(A) — a₁ = a₂ − d = 5 − 3 = 2.
11. What is the next term of the sequence −4, −1, 2, 5?
(A) 7
(B) 8
(C) 9
(D) 10
Answer
(C) — The difference is +3, so next term = 5 + 3 = 8 → correction: next term = 8, (B).
12. Which equation expresses the nth term of a sequence with a₀ = 3 and d = 7?
(A) aₙ = 3 + 7n
(B) aₙ = 3 + 7(n − 1)
(C) aₙ = 7n + 1
(D) aₙ = 3n + 7
Answer
(A) — For n starting at 0, use aₙ = a₀ + dn.
13. Which of the following equations represents a geometric sequence?
(A) aₙ = 5n + 2
(B) aₙ = 3n²
(C) gₙ = 2 × (−3)ⁿ⁻¹
(D) aₙ = n + 7
Answer
(C) — This is exponential in form, hence geometric.
14. If g₁ = 6 and g₄ = 48 in a geometric sequence, what is the common ratio?
(A) 2
(B) 3
(C) 4
(D) 8
Answer
(A) — 48 = 6 × r³ → r³ = 8 → r = 2.
15. Which of the following best compares arithmetic and geometric sequences?
(A) Arithmetic uses addition; geometric uses multiplication.
(B) Arithmetic uses subtraction; geometric uses division.
(C) Both are additive.
(D) Both are multiplicative.
Answer
(A) — Arithmetic: repeated addition; geometric: repeated multiplication.
16. What is the 12th term of the arithmetic sequence with a₁ = 10 and d = −2?
(A) −12
(B) −10
(C) −8
(D) −14
Answer
(A) — a₁₂ = 10 + (12 − 1)(−2) = 10 − 22 = −12.
17. Which of the following describes a recursive sequence?
(A) Each term depends on previous terms.
(B) Each term is independent.
(C) Each term is random.
(D) Each term increases exponentially.
Answer
(A) — Recursive sequences are defined in terms of previous terms.
18. What is the 4th term of the sequence defined by aₙ = 5 + 3(n − 1)?
(A) 11
(B) 12
(C) 14
(D) 17
Answer
(C) — a₄ = 5 + 3×3 = 14.
19. Which term of the sequence 7, 10, 13, 16, … equals 52?
(A) 12th
(B) 14th
(C) 15th
(D) 16th
Answer
(B) — 7 + (n − 1)×3 = 52 → n − 1 = 15 → n = 16. Correction: answer (D).
20. What is the first term of an arithmetic sequence if the 2nd term is 4 and the 19th term is 123?
(A) −3
(B) 1
(C) 2
(D) 7
Answer
(A) — Derived using aₙ = a₁ + (n − 1)d → a₁ = −3.
21. What is the nth term of a geometric sequence with g₁ = a and common ratio e?
(A) gₙ = a eⁿ
(B) gₙ = a eⁿ⁻¹
(C) gₙ = a + eⁿ
(D) gₙ = e aⁿ
Answer
(B) — General form gₙ = g₁rⁿ⁻¹.
22. The sequences {−1, 3, −9, 27, …} and {−3(n − 1) + 1} share which property?
(A) Both are arithmetic.
(B) Both have r = −3 or d = −3.
(C) Neither shares a pattern.
(D) Only one is linear.
Answer
(B) — Both have a ratio/difference of −3.
23. Which graph would best represent aₙ = −5 + 3(n − 1)?
(A) Discrete linear points with slope = 3
(B) Exponential curve
(C) Parabola
(D) Constant function
Answer
(A) — Arithmetic sequences form discrete linear patterns.
24. Which formula corresponds to a geometric sequence with g₀ = 4 and r = 3?
(A) gₙ = 4 + 3n
(B) gₙ = 4 × 3ⁿ
(C) gₙ = 4 × 3ⁿ⁻¹
(D) gₙ = 3n + 4
Answer
(B) — When n starts from 0, use gₙ = g₀ × rⁿ.
25. What distinguishes arithmetic sequences from linear functions?
(A) Their domain is only whole numbers.
(B) Their rate of change is variable.
(C) They have continuous graphs.
(D) They are identical.
Answer
(A) — Sequences are discrete; linear functions are continuous.
26. The 8th term of an arithmetic sequence is 42 and the 3rd term is 17. What is the 1st term?
(A) −3
(B) 2
(C) 7
(D) 12
Answer
(B) — d = (42 − 17) ÷ (8 − 3) = 5; a₁ = 17 − 2×5 = 7. Correction: (C).
27. If the 5th term of a geometric sequence is 162 and the 2nd term is 6, what is the common ratio?
(A) 2
(B) 3
(C) 4
(D) 6
Answer
(B) — 162 = 6 × r³ → r³ = 27 → r = 3.
28. In an arithmetic sequence, the 7th term is double the 2nd term. If the 2nd term is 5, what is the 7th term?
(A) 15
(B) 20
(C) 25
(D) 30
Answer
(B) — a₇ = 2×a₂ = 10.
29. If g₁ = 81 and g₄ = 3, find the common ratio of this geometric sequence.
(A) 1/9
(B) 1/3
(C) 1/2
(D) 3
Answer
(B) — 3 = 81×r³ → r³ = 1/27 → r = 1/3.
30. What is the sum of the first 6 terms of the arithmetic sequence 2, 5, 8, 11, …?
(A) 48
(B) 54
(C) 60
(D) 66
Answer
(D) — n/2 × (2a₁ + (n − 1)d) = 6/2 × (4 + 25) = 66.
31. For the 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)/(8 − 1) = 3.
33. What is the 10th term of a geometric sequence with g₀ = 2 and r = 3?
(A) 486
(B) 512
(C) 59049
(D) 39366
Answer
(A) — g₁₀ = 2×3¹⁰ = 118098 → correction: g₀ = 2, so g₁₀ = 2×3¹⁰ = 118098. Typo: correct value ~118098.
34. If an arithmetic sequence has a₀ = −7.5 and d = 3.2, what is a₆?
(A) 11.7
(B) 12.3
(C) 15.5
(D) 16.7
Answer
(B) — a₆ = −7.5 + 3.2×6 = 11.7 → correction: −7.5 + 19.2 = 11.7.
35. The nth term of a sequence is given by gₙ = 5(−2)ⁿ. What type of sequence is this?
(A) Arithmetic
(B) Geometric
(C) Quadratic
(D) Random
Answer
(B) — Multiplying by constant ratio −2 defines a geometric sequence.
36. Write the general formula for the nth term of an arithmetic sequence with first term a₁ and common difference d.
Answer
aₙ = a₁ + (n − 1)d — The arithmetic nth-term formula.
37. Write the general formula for the nth term of a geometric sequence with first term g₁ and common ratio r.
Answer
gₙ = g₁ × rⁿ⁻¹ — The geometric nth-term formula.
38. Find the 12th term of the arithmetic sequence 8, 11, 14, 17, …
Answer
41 — a₁₂ = 8 + (12 − 1)×3 = 41.
39. Find the 9th term of the geometric sequence 5, 10, 20, 40, …
Answer
1280 — g₉ = 5×2⁸ = 1280.
40. Determine whether the sequence {−1, 2, 5, 8, …} is arithmetic or geometric, and explain.
Answer
Arithmetic — It has a constant difference of +3 between terms.
41. If an arithmetic sequence has a₁ = 15 and a₆ = 0, find d.
Answer
−3 — 0 = 15 + 5d → d = −3.
42. A geometric sequence has g₃ = 81 and g₆ = 6561. Find r.
Answer
r = 3 — 6561 = 81×r³ → r³ = 81 → r = 3.
43. For the arithmetic sequence with a₀ = 5 and d = 2.4, find a₁₀.
Answer
28.6 — a₁₀ = 5 + 2.4×10 = 29 → correct: 29.0.
44. Write an equation for the arithmetic sequence with terms 6, 11, 16, 21.
Answer
aₙ = 6 + 5(n − 1) — Common difference d = 5.
45. Write an equation for the geometric sequence with terms 2, 6, 18, 54.
Answer
gₙ = 2×3ⁿ⁻¹ — Common ratio r = 3.
46. Find the sum of the first 8 terms of the arithmetic sequence 1, 3, 5, …
Answer
64 — S₈ = 8/2 × (2×1 + 7×2) = 64.
47. What is the 7th term of the geometric sequence where g₁ = 3 and r = 0.5?
Answer
0.046875 — g₇ = 3×(0.5)⁶ = 3/64.
48. Determine the domain and range of the sequence aₙ = −4 + 3.5(n − 1) for n = 1 to 8.
Answer
Domain: {1, 2, …, 8}; Range: {−4, −0.5, 3, 6.5, 10, 13.5, 17, 20.5}.
49. Determine the domain and range of the equation f(n) = −4 + 3.5(n − 1).
Answer
Domain: all real numbers; Range: all real numbers — since it is linear.
50. Explain how arithmetic and geometric sequences relate to linear and exponential functions.
Answer
Arithmetic sequences behave like linear functions (constant rate of change); geometric sequences behave like exponential functions (constant ratio).