Difference between revisions of "1951 AHSME Problems/Problem 11"
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Revision as of 08:56, 9 December 2011
Problem
The limit of the sum of an infinite number of terms in a geometric progression is $\frac {a}{1 \minus{} r}$ (Error compiling LaTeX. Unknown error_msg) where 
denotes the first term and $\minus{} 1 < r < 1$ (Error compiling LaTeX. Unknown error_msg) denotes the common ratio. The limit of the sum of their squares is:
$\textbf{(A)}\ \frac {a^2}{(1 \minus{} r)^2} \qquad\textbf{(B)}\ \frac {a^2}{1 \plus{} r^2} \qquad\textbf{(C)}\ \frac {a^2}{1 \minus{} r^2} \qquad\textbf{(D)}\ \frac {4a^2}{1 \plus{} r^2} \qquad\textbf{(E)}\ \text{none of these}$ (Error compiling LaTeX. Unknown error_msg)
Solution
Let the original geometric series be 
. Therefore, their squares are 
, which is a geometric sequence with first term 
and common ratio 
. Thus, the sum is $\boxed{\textbf{(C)}\ \frac {a^2}{1 \minus{} r^2}}$ (Error compiling LaTeX. Unknown error_msg).
See also
| 1951 AHSME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 10 |
Followed by Problem 12 | |
| 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 | ||
