1952 AHSME Problem 50

A line initially 1 inch long grows according to the following law, where the first term is the initial length. 1+\frac{1}{4}\sqrt{2}+\frac{1}{4}+\frac{1}{16}\sqrt{2}+\frac{1}{16}+\frac{1}{64}\sqrt{2}+\frac{1}{64}+\cdots

If the growth process continues forever, the limit of the length of the line is:

\textbf{(A) } \infty\qquad \textbf{(B) } \frac{4}{3}\qquad \textbf{(C) } \frac{8}{3}\qquad \textbf{(D) } \frac{1}{3}(4+\sqrt{2})\qquad \textbf{(E) } \frac{2}{3}(4+\sqrt{2})

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