Answer
See below.
Work Step by Step
Proofs using mathematical induction consists of two steps:
1) The base case: here we prove that the statement holds for the first natural number.
2) The inductive step: assume the statement is true for some arbitrary natural number $n$ larger than the first natural number, then we prove that then the statement also holds for $n + 1$.
Hence here:
1) For $n=1: 2=\frac{1(5(1)-1)}{2}$.
2) Assume for $n=k: 2+7+...+5k-3=\frac{k(5(k)-1)}{2}$. Then for $n=k+1$:
$2+7+...+5k-3+5k+2=\frac{k(5(k)-1)}{2}+5k+2=\frac{5k^2-5k}{2}+\frac{10k+4}{2}=\frac{5k^2+5k+4}{2}=\frac{(k+1)((5k+4)}{2}=\frac{(k+1)((5(k+1)-1)}{2}.$
Thus we proved what we wanted to.