gaokao Mathematics: Sequences and Series 2 — Questions and Answers
Question 1: In an arithmetic sequence, a₃ = 7 and a₇ = 15. What is a₁?
- 2
- 3 (Correct answer)
- 4
- 5
Correct answer: 3
From a₃ = a₁ + 2d = 7 and a₇ = a₁ + 6d = 15, subtracting gives 4d = 8, d = 2, so a₁ = 7 − 4 = 3.
Question 2: The sum of the first n terms of a sequence is Sₙ = n² + 2n. What is a₅?
- 9
- 11 (Correct answer)
- 13
- 10
Correct answer: 11
For n ≥ 2, aₙ = Sₙ − Sₙ₋₁ = (n² + 2n) − ((n−1)² + 2(n−1)) = 2n + 1; so a₅ = 2(5) + 1 = 11.
Question 3: In a geometric sequence, b₂ = 6 and b₅ = 48. What is the common ratio?
- 2 (Correct answer)
- 3
- 4
- 8
Correct answer: 2
b₅/b₂ = q³ = 48/6 = 8, so q = ∛8 = 2.
Question 4: The sum of the first 5 terms of an arithmetic sequence is 25, and the sum of the first 10 terms is 75. What is the common difference d?
- 0
- 1 (Correct answer)
- 2
- 3
Correct answer: 1
S₅ = 5a₁ + 10d = 25 gives a₁ + 2d = 5; S₁₀ = 10a₁ + 45d = 75 gives 2a₁ + 9d = 15; solving yields d = 1.
Question 5: What is the value of ∑(k=1 to 5) (2k − 1)?
- 20
- 25 (Correct answer)
- 30
- 35
Correct answer: 25
The sum is 1 + 3 + 5 + 7 + 9 = 25; this is also the sum of the first 5 odd numbers, which equals 5² = 25.
Question 6: A sequence is defined recursively by a₁ = 1 and aₙ₊₁ = aₙ + 3. What is a₈?
- 20
- 21
- 22 (Correct answer)
- 24
Correct answer: 22
This is arithmetic with d = 3; using a₈ = a₁ + 7d = 1 + 7(3) = 1 + 21 = 22.
Question 7: An arithmetic sequence has a₁ = 5 and a₂₀ = 62. What is the sum S₂₀?
- 630
- 650
- 670 (Correct answer)
- 690
Correct answer: 670
Using Sₙ = n(a₁ + aₙ)/2, S₂₀ = 20(5 + 62)/2 = 10 × 67 = 670.
gaokao Mathematics: Sequences and Series 2
In an arithmetic sequence, a₃ = 7 and a₇ = 15.
What is a₁?