2009 AMC 12B Problems/Problem 12

Problem

The fifth and eighth terms of a geometric sequence of real numbers are $7!$ and $8!$ respectively. What is the first term?

$\mathrm{(A)}\ 60\qquad \mathrm{(B)}\ 75\qquad \mathrm{(C)}\ 120\qquad \mathrm{(D)}\ 225\qquad \mathrm{(E)}\ 315$

Solution

Let the $n$ th term of the series be $ar^{n-1}$ . Because $\frac {8!}{7!} = \frac {ar^7}{ar^4} = r^3 = 8,$ it follows that $r = 2$ and the first term is $a = \frac {7!}{r^4} = \frac {7!}{16} = \boxed{315}$ . The answer is $\mathrm{(E)}$ .

2009 AMC 12B (Problems • Answer Key • Resources)
Preceded by Problem 11	Followed by Problem 13
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2009 AMC 12B Problems/Problem 12

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