Difference between revisions of "2007 AMC 12B Problems/Problem 15"

Revision as of 12:31, 23 December 2016

Problem 15

The geometric series $a+ar+ar^2\ldots$ has a sum of $7$ , and the terms involving odd powers of $r$ have a sum of $3$ . What is $a+r$ ?

$\textbf{(A)}\ \frac{4}{3}\qquad\textbf{(B)}\ \frac{12}{7}\qquad\textbf{(C)}\ \frac{3}{2}\qquad\textbf{(D)}\ \frac{7}{3}\qquad\textbf{(E)}\ \frac{5}{2}$

Solution

Solution 1

The sum of an infinite geometric series is given by $\frac{a}{1-r}$ where $a$ is the first term and $r$ is the common ratio.

In this series, $\frac{a}{1-r} = 7$

The series with odd powers of $r$ is given as $ar + ar^3 + ar^5 ...$

It's sum can be given by $\frac{ar}{1-r^2} = 3$

Doing a little algebra

$ar = 3(1-r)(1+r)$

$ar = 3\left(\frac{a}{7}\right)(1+r)$

$\frac{7}{3}r = 1 + r$

$r = \frac{3}{4}$

$a = 7(1-r) = \frac{7}{4}$

$a + r =\boxed{ \frac{5}{2}} \Rightarrow \mathrm{(E)}$

Solution 2

The given series can be decomposed as follows: $(a + ar + ar^2 + \ldots) = (a + ar^2 + ar^4 + \ldots) + (ar + ar^3 + ar^5 + \ldots)$

Clearly $(a + ar^2 + ar^4 + \ldots) = (ar + ar^3 + ar^5 + \ldots)/r = 3/r$ . We obtain that $7 = 3/r + 3$ , hence $r = \frac{3}{4}$ .

Then from $7 = (a + ar + ar^2 + \ldots) = \frac{a}{1-r}$ we get $a=\frac{7}{4}$ , and thus $a + r = \frac{5}{2}$ .

2007 AMC 12B (Problems • Answer Key • Resources)
Preceded by Problem 14	Followed by Problem 16
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All AMC 12 Problems and Solutions

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

@@ Line 26: / Line 26: @@
 <math>a = 7(1-r) = \frac{7}{4}</math>
-<math>a + r = \frac{5}{2} \Rightarrow \mathrm{(E)}</math>
+<math>a + r =\boxed{ \frac{5}{2}} \Rightarrow \mathrm{(E)}</math>
 ===Solution 2===

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