Suppose $~a_0 = 1~$ and $~a_n = 2a_{n−1} + 3~$ for all $~n ≥ 1$. Using proof by induction, prove that the formula for an for all $~n ≥ 0~$ is given by $~a_n = 2^{n+2} − 3$.
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Hint: Show that the case for $n=0$ holds. Then, assuming the formula holds for $n=k$, show that the formula for $n=k+1$ holds. This will likely require you to use the actual recurrence to express $a_{k+1}$ in terms of $a_k$. If you want some more detailed help, please [edit] your post to include some details on your own attempts and understanding of this problem. – PrincessEev Nov 21 '19 at 06:42