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:
1) For $n=1: 1=\frac{1}{4}(1+3)$
2) Assume for $n=k: 1+1.5+...+0.5(k+1)=\frac{k}{4}(k+3)$.
Then $n=k+1:1+1.5+...+0.5(k+1)+0.5(k+2)=\frac{k}{4}(k+3)+0.5(k+2)=0.25k^2+0.75k+0.5k+1=0.25(k+1)(k+4)=0.25(k+1)((k+1)+3)$
Thus we proved what we wanted to.
