Question: 2022 Winter Final - 10
The function $ T: \mathbb{N} \rightarrow \mathbb{N} $ is defined recursively as follows:
\[
T(n) =
\begin{cases}
3 & \text{if $n=0$} \\
2\cdot T(n-1) + 3 & \text{if $n\ge 1$}
\end{cases}
\]
Which of the following is true, for all $n\ge 0$?
(a)
$T(n)=3(2^{n+1}-1)$
(b)
$T(n)= 3n$
(c)
$T(n)=2(3^{n+1}-1)$
(d)
$T(n)= 3^{n+1}$
(e)
$T(n)= 2^{n+1}+1$