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Math Help - induction

  1. #1
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    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.
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  2. #2
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    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 \\ <br />
 & =2m^{2}+7m+5+4(m+1)+5 \\ <br />
 & =2(m+1)^{2}+7(m+1)+5.\quad\blacksquare<br />
\end{aligned}
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