1952 AMC 12 Problem 50

Question

A line initially

1

inch long grows according to the following law, where the first term is the initial length.

\begin{gathered} 1+\frac14\sqrt2+\frac14+\frac1{16}\sqrt2\\ {}+\frac1{16}+\frac1{64}\sqrt2+\frac1{64}+\cdots. \end{gathered}

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

\infty

\dfrac43

\dfrac38

\dfrac13(4+\sqrt2)

\dfrac23(4+\sqrt2)

Group each pair having the same power of

\frac{1}{4}

The terms after the initial

1

equal

(1+\sqrt2)\sum_{k=1}^{\infty}4^{-k}

After the initial term, each power

4^{-k}

appears once by itself and once multiplied by

\sqrt2.

Since

\sum_{k=1}^{\infty}4^{-k}=\frac{1}{3},

the limit is

1+\frac{1+\sqrt2}{3}=\frac{4+\sqrt2}{3}.

Thus, the correct answer is D.

Problem 50 in other years