Answer
See below.
Work Step by Step
Proofs using mathematical induction consist 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 we have:
1) For $n=1: 5=1(3(1)+7)/2$.
2) Assume for $n=k: 5+8++3k+2=\frac{k(3k+7)}{2}$. Then for $n=k+1$:
$5+8++3k+2+3(k+1)+2=\frac{k(3k+7)}{2}+3(k+1)+2=\frac{k(3k+7)}{2}+3k+5=\frac{(k+1)(3k+10)}{2}=\frac{(k+1)(3(k+1)+7)}{2}.$
Thus we proved what we wanted to.