1974 AMC 12 Problem 21

Attempt Problem 21 of the 1974 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 1974 AMC 12 solutions, or check the answer key.

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21.

In a geometric series of positive terms the difference between the fifth and fourth terms is

576,

and the difference between the second and first terms is

9.

What is the sum of the first five terms of this series?

1061

1023

1024

768

none of these

Write the first term as

a

and the common ratio as

r

Divide

ar^3(r-1)=576

a(r-1)=9

Let the first term be

a

and the ratio be

r.

The conditions give

\begin{aligned} ar^3(r-1)&=576,\\ a(r-1)&=9. \end{aligned}

Their quotient gives

r^3=64,

r=4.

Then

3a=9,

hence

a=3.

The sum is

\begin{aligned} S&=3(1+4+16+64+256)\\ &=3\cdot341\\ &=1023. \end{aligned}

Therefore, the correct answer is B.

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