2016 AMC 12B Problem 21

Let ABCD be a unit square. Let Q_1 be the midpoint of \overline{CD}. For i=1,2,\ldots, let P_i be the intersection of \overline{AQ_i} and \overline{BD}, and let Q_{i+1} be the foot of the perpendicular from P_i to \overline{CD}. What is \sum_{i=1}^{\infty} \text{Area of } \triangle DQ_i P_i \, ?

\textbf{(A)}\ \frac{1}{6} \qquad \textbf{(B)}\ \frac{1}{4} \qquad \textbf{(C)}\ \frac{1}{3} \qquad \textbf{(D)}\ \frac{1}{2} \qquad \textbf{(E)}\ 1

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