Question: How to derive this formula? 

$$n^2=\sum\_{k=1}^n(2k-1).$$

Answer: 
*[Notice](http://en.wikipedia.org/wiki/Square_number#Properties) how the difference of two consecutive squares is always an odd number:* $$(x+1)^2=x^2+(2x+1)$$
[![Image](https://i.stack.imgur.com/CJiZi.png)](http://betterexplained.com/articles/surprising-patterns-in-the-square-numbers-1-4-9-16/)
Hint:

$$2(1)-1+2(2)-1+2(3)-1+\dots 2(n)-1$$
$$=2(1)+2(2)+2(3)+\dots+2(n)-(1+1+\dots 1)$$

$$=2(1+2+3+\dots+n)-n$$

At least now you should be able to do If you are familiar with the formula for :

> 
> sum of first $n$ natural numbers
>  = ??
> 
> 
>