Question: 2018 Fall Final - 10
The function $T : \mathbb{N} \rightarrow \mathbb{N}$ is recursively defined as follows:
$$
\begin{align}
T(0) &= 2, \\
T(n) &= 3 \cdot T(n - 1) + 1,\ \ \mathrm{if}\ n \geq 1.
\end{align}
$$
Which of the following is true for all integers $n \geq 0$?
(a)
$T(n) = \frac{5}{2} \cdot 3^{n} - \frac{1}{2}$
(b)
$T(n) = \frac{3}{2} \cdot 3^{n} - \frac{1}{2}$
(c)
None of the above.
(d)
$T(n) = \frac{5}{2} \cdot 3^{n} - 1$