Question 50
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:
\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: