[Solved]: Induction proof, base case not working but induction step works?

Problem Detail: 

$1+3+5+...+(2n+3)=n^2+4n$

For this series using induction proof.

Base case 1,2,3,.. not working. But induction step works well.

Base case is not given in question.

Asked By : user5507

Answered By : yuanshang

First of all, the right hand side should be $n^2+4n+4$

You can find the generalized term for the left hand side is $2n-1$. Therefore,

$1+3+5+...+(2n-1)+(2n+1)+(2n+3)=n^2+4n+4$, which gives you:

$1+3+5+...+(2n-1)=n^2$.

Then you can apply the mathematical induction to prove it holds for every positive integer $n$.

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Question Source : http://cs.stackexchange.com/questions/21593