$2+4+6+\cdots+2n=n^2+n$ by mathematical induction

Question

I am trying to prove $2+4+6+\cdots +2n=n^2+n$ by mathematical induction. I followed all the steps and the $P_{k+1}$ was $2+4+6+\cdots+2(k+1)=(k+1)^2+k+1$ Starting from the left hand side of the equation I have solved till $k^2+k+2(k+1)$. Now I am stuck here. I don't know how to do it further. Please guide me thanks.

Answer 1

It looks as if you need $2+4+6+\cdots+2n$ rather than $2+4+6+\cdots+2^n$.

Look at $$ \underbrace{2+4+6+\cdots+2k}+2(k+1). $$ The induction hypothesis tells you what to do with the part over the underbrace. Then massage it a bit with some simple algebra.

Answer 2

$k^2+k+2(k+1)=k^2+3k+2=k^2+2k+1+(k+1)= (k+1)^2+(k+1)$