AMC 12 AMC 12 Sequences, Series & Limits

Question 1

What is the sum of the arithmetic series 1 + 2 + 3 + … + 100?

Accepted Answer

5050

Answer

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, …?

Accepted Answer

2

Answer

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 + …?

Accepted Answer

2

Answer

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?

Accepted Answer

32

Answer

aₙ = a₁ + (n − 1)d = 5 + 9·3 = 32.

Question 5

What is the nth term of the geometric sequence 2, 6, 18, 54, …?

Accepted Answer

2·3^(n−1)

Answer

With a₁ = 2 and r = 3, the nth term is a₁·r^(n−1) = 2·3^(n−1).

AMC 12 - - American Mathematics Contest 12 Practice Test