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:

$\infty$

$\dfrac43$

$\dfrac38$

$\dfrac13(4+\sqrt2)$

$\dfrac23(4+\sqrt2)$

Answer: D

Concepts:geometric sequence summation radical

Difficulty rating: 1640

Small Hint:

Group each pair having the same power of $\frac{1}{4}$

Big Hint:

The terms after the initial $1$ equal $(1+\sqrt2)\sum_{k=1}^{\infty}4^{-k}$

Solution:

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.

1952 AMC 12 Problem 50

Answer: D

Difficulty rating: 1640

Small Hint:

Big Hint:

Solution:

Problem 50 in Other Years