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