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 $n=1: (1+3)^2=16\lt2(3^2)=18$
2) Assume for $n=k: (k+1)^2\lt2k^2$. Then for $n=k+1$:
$(k+1+1)^2=k^2+4k+1\lt=2k^2+4k+2=2(k+1)^2$
Thus we proved what we wanted to.