# 1974 AHSME Problems/Problem 21

## Problem

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? $\mathrm{(A)\ } 1061 \qquad \mathrm{(B) \ }1023 \qquad \mathrm{(C) \ } 1024 \qquad \mathrm{(D) \ } 768 \qquad \mathrm{(E) \ }\text{none of these}$

## Solution

Let the first term be $a$ and the common ratio be $r$. Therefore, the second term is $ar$, the fourth term is $ar^3$, and the fifth term is $ar^4$. We're given that $ar^4-ar^3=576\implies ar^3(r-1)=576$ and $ar-a=9\implies a(r-1)=9$. Dividing this first equation by this second one, we get $r^3=\frac{576}{9}=64\implies r=4$. Therefore, $a(4-1)=9$, so $a=3$.

Therefore, the first five terms of this series are $3, 12, 48, 192, 768$, and their sum is $1023, \boxed{\text{B}}$.

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