1951 AHSME Problems/Problem 11

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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. ! Undefined control sequence.) where $a$ denotes the first term and $\minus{} 1 < r < 1$ (Error compiling LaTeX. ! Undefined control sequence.) 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. ! Undefined control sequence.)

Solution

Let the original geometric series be $a,ar,ar^2,ar^3,ar^4\cdots$. Therefore, their squares are $a^2,a^2r^2,a^2r^4,a^2r^6,\cdots$, which is a geometric sequence with first term $a^2$ and common ratio $r^2$. Thus, the sum is $\boxed{\textbf{(C)}\ \frac {a^2}{1 \minus{} r^2}}$ (Error compiling LaTeX. ! Undefined control sequence.).

See also

1951 AHSME (ProblemsAnswer KeyResources)
Preceded by
Problem 10
Followed by
Problem 12
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