# 1974 AHSME Problems/Problem 21

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## 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 $1023, \boxed{\text{B}}$.

## See Also

 1974 AHSME (Problems • Answer Key • Resources) Preceded byProblem 20 Followed byProblem 22 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 All AHSME Problems and Solutions

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