Answer
See below
Work Step by Step
1. For $n=3$, we have $LHS=4^3=64$ and $RHS=\frac{4(4^3-16)}{3}=64=LHS$, thus P(3) is true.
2. Assume P(k) ($k\gt3$) is true, we have:
$4^3+4^4+...+4^k=\frac{4(4^k-16)}{3}$
3. For $n=k+1$, we have:
$LHS=4^3+4^4+...+4^k+4^{k+1}=\frac{4(4^k-16)}{3}+4^{k+1}\\
=\frac{4^{k+1}-4(16)+3(4^{k+1})}{3}=\frac{4(4^{k+1}-16)}{3}=RHS$
4. Thus $P(k+1)$ is also true and we have proved the statement by mathematical induction.