[Math] Induction proof of a Recurrence Relation

Question

Consider the following recurrence equation obtained from a recursive algorithm:

Using Induction on n, prove that:

So I got my way thru step1 and step2: the base case and hypothesis step but I'm not sure how to proceed. please help

Answer 1

Base Case: $n = 1$
$\quad T(1) = 2^{1+1}-1 = 3$

Inductive Hypothesis:
$\quad$ Assume $T(n)=2^{n+1}-1$ is true for some $n \ge 1$

Inductive Step: $n+1$ (since $n \ge 1,\; (n+1) \ge 2$)
$\quad T(n+1) = T(n) + 2^{n+1} \quad\quad\quad$ (by recurrence relation)
$\quad\quad\quad\quad\quad = 2^{n+1} - 1 + 2^{n+1} \quad\;\;$ (by inductive hypothesis)
$\quad\quad\quad\quad\quad = 2^{(n+1)+1}-1$

which proves the case for n+1