Answer
See below.
Work Step by Step
1. Let $P(n)$ be the statement to be proved.
2. For $n=0$, we have $7^{0}-2^0=0$ and $0|5$, thus $P(0)$ is true.
3. Assume $P(k), k\gt0$ is true, that is $7^{k}-2^k$ is divisible by 5.
4. For $n=k+1$, we have $7^{k+1}-2^{k+1}=7\cdot7^k-2\cdot2^k=7\cdot7^k-7\cdot2^k+5\cdot2^k=7(7^{k}-2^k)+5\cdot2^k$
which is divisible by 5.
5. Thus $P(k+1)$ is also true and we have proved the statement by mathematical induction.