1951 AHSME Problem 11

The limit of the sum of an infinite number of terms in a geometric progression is \frac {a}{1- r} where a denotes the first term and -1 < r < 1 denotes the common ratio. The limit of the sum of their squares is:

\textbf{(A)}\ \frac {a^2}{(1 -r)^2} \qquad\textbf{(B)}\ \frac {a^2}{1 + r^2} \qquad\textbf{(C)}\ \frac {a^2}{1 - r^2} \qquad\textbf{(D)}\ \frac {4a^2}{1+ r^2} \qquad\textbf{(E)}\ \text{none of these}

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.