Question: 2019 Winter Final - 10
The function $f : \mathbb{N} \rightarrow \mathbb{N}$ is recursively defined as follows:
$$
\begin{align}
f(0) &= 2 \\
f(n) &= 3 \cdot f(n - 1) + 1, \text{ if } n \geq 1
\end{align}
$$
Which of the following is true for all integers $n \geq 0$?
(a)
$f(n) = \frac{5}{2} \cdot 3^n - 1$
(b)
$f(n) = \frac{3}{2} \cdot 3^n - \frac{1}{2}$
(c)
$f(n) = \frac{5}{2} \cdot 3^n - \frac{1}{2}$
(d)
None of the above.