1975 AMC 12 Problem 16

Attempt Problem 16 of the 1975 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1975 AMC 12 solutions, or check the answer key.

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16.

If the first term of an infinite geometric series is a positive integer, the common ratio is the reciprocal of a positive integer, and the sum of the series is

3,

then the sum of the first two terms of the series is

\frac13

\frac23

\frac83

2

\frac92

Answer: C

Concepts:geometric sequence divisibility summation

Difficulty rating: 1740

Small hint:

Write the first term as

a

and the ratio as

\frac{1}{n}

Big hint:

From

\frac{a}{1-\frac{1}{n}}=3,

use that

a,n

are positive integers and

n\gt1

Solution:

Let the first term be

a

and the ratio

\frac{1}{n}.

Convergence gives

n\gt1,

and

\frac{a}{1-\frac{1}{n}}=3,\qquad a=3-\frac3n.

Thus

n

divides

3,

n=3

and

a=2.

The first two terms sum to

2+\frac{2}{3}=\frac{8}{3}.

Therefore, the correct answer is C.