AMC12 Sequences and Series

Question 1

What is the 10th term of the arithmetic sequence 3, 7, 11, 15, ...?

Accepted Answer

39

Answer

The common difference is 4, so a₁₀ = 3 + 9 × 4 = 3 + 36 = 39.

Question 2

What is the sum of the first 100 positive integers?

Accepted Answer

5050

Answer

Sum = n(n+1)/2 = 100 × 101/2 = 5050, as computed by Gauss.

Question 3

A geometric sequence has first term 2 and common ratio 3. What is the 5th term?

Accepted Answer

162

Answer

a₅ = a₁ × r^(n−1) = 2 × 3⁴ = 2 × 81 = 162.

Question 4

What is the sum of the infinite geometric series 1 + 1/2 + 1/4 + 1/8 + ...?

Accepted Answer

2

Answer

S∞ = a/(1−r) = 1/(1−1/2) = 2.

Question 5

In the Fibonacci sequence 1, 1, 2, 3, 5, 8, ..., what is the 8th term?

Accepted Answer

21

Answer

Continuing the Fibonacci sequence: the 8th term is 21.

AMC12 - American Mathematics Competition 12 Practice Test