AMC 12 AMC 12 Sequences, Series & Limits 1 — Questions and Answers
Question 1: What is the sum of the arithmetic series 1 + 2 + 3 + … + 100?
- 100
- 1000
- 5050 (Correct answer)
- 10100
Correct answer: 5050
The sum of the first n positive integers is n(n+1)/2 = 100·101/2 = 5050.
Question 2: What is the common ratio of the geometric sequence 3, 6, 12, 24, …?
- 3
- 2 (Correct answer)
- 6
- 1/2
Correct answer: 2
Each term is twice the previous term, so the common ratio r = 6/3 = 2.
Question 3: What is the sum of the infinite geometric series 1 + 1/2 + 1/4 + 1/8 + …?
- 1
- 2 (Correct answer)
- 3
- ∞
Correct answer: 2
For |r| < 1, the sum is a/(1 − r) = 1/(1 − 1/2) = 2.
Question 4: In an arithmetic sequence with a₁ = 5 and common difference d = 3, what is a₁₀?
- 30
- 32 (Correct answer)
- 35
- 27
Correct answer: 32
aₙ = a₁ + (n − 1)d = 5 + 9·3 = 32.
Question 5: What is the nth term of the geometric sequence 2, 6, 18, 54, …?
- 2·3^n
- 2·3^(n−1) (Correct answer)
- 3·2^(n−1)
- 6^(n−1)
Correct answer: 2·3^(n−1)
With a₁ = 2 and r = 3, the nth term is a₁·r^(n−1) = 2·3^(n−1).
Question 6: What is the sum of the first 5 terms of a geometric series with a₁ = 4 and r = 2?
- 60
- 124 (Correct answer)
- 96
- 32
Correct answer: 124
Sₙ = a₁(rⁿ − 1)/(r − 1) = 4(2⁵ − 1)/(2 − 1) = 4·31 = 124.
AMC 12 AMC 12 Sequences, Series & Limits 1
What is the sum of the arithmetic series 1 + 2 + 3 + … + 100?