Question: 2017 Winter Midterm - 12
The function $f : \{1,2,3,\dots\} \rightarrow \mathbb{R}$ is defined by
$$
\begin{align}
f(1) &= 2, \\
f(n) &= 2 \left( \frac{n - 1}{n} \right)^2 \cdot f(n - 1)\ \mathrm{if}\ n \geq 2.
\end{align}
$$
What is $f(n)$?
(a)
$f(n) = \frac{2^{n-1}}{n^{2}}$
(b)
$f(n) = \frac{2^{n}}{n^{2}}$
(c)
$f(n) = \frac{2^{n}}{(n-1)^{2}}$
(d)
$f(n) = \frac{n^{2}}{2^{n}}$