# Math Help - induction

1. ## induction

Looking for some help with this problem, Thanks ahead of time

Write out an induction proof of the following equation for all n N.
5 + 9 + 13 + ... + (4n + 5) = 2n^2 + 7n + 5.

2. Base case: $n=1,$ so $\sum\limits_{k\,=\,0}^{1}{(4k+5)}=5+9=14$ besides $2\cdot 1^{2}+7\cdot 1+5=14.$ Now asume valid the proposition for some $n=m,$ our I.H. will be $\sum\limits_{k\,=\,0}^{m}{(4k+5)}=2m^{2}+7m+5.$ It remains to prove that $\sum\limits_{k\,=\,0}^{m+1}{(4k+5)}=2(m+1)^{2}+7(m +1)+5.$ Observe that:

\begin{aligned}\sum\limits_{k\,=\,0}^{m+1}{(4k+5)} &=\underbrace{\sum\limits_{k\,=\,0}^{m}{(4k+5)}}_{ \text{I}\text{.H}\text{.}}+\,4(m+1)+5 \\
& =2m^{2}+7m+5+4(m+1)+5 \\