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 $n=1: 3=1(2(1)+1)$.
2) Assume for $n=k: 3+7+...+(4k-1)=k(2k+1)$. Then for $n=k+1$:
$3+7+...+(4k-1)+4k+3=k(2k+1)+4k+3=2k^2+k+4k+31=(k+1)(2k+3)=(k+1)(2(k+1)+1).$
Thus we proved what we wanted to.
