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

Revision as of 09:20, 30 May 2012

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
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

@@ Line 11: / Line 11: @@
 ==See Also==
 {{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"

Revision as of 09:20, 30 May 2012

Problem

Solution

See Also