Difference between revisions of "1974 AHSME Problems/Problem 21"

Latest revision as of 11:43, 5 July 2013

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

1974 AHSME (Problems • Answer Key • Resources)
Preceded by Problem 20	Followed by Problem 22
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All AHSME Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

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 {{AHSME box|year=1974|num-b=20|num-a=22}}
 [[Category:Introductory Algebra Problems]]
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Difference between revisions of "1974 AHSME Problems/Problem 21"

Latest revision as of 11:43, 5 July 2013

Problem

Solution

See Also