Prove that the sum of consecutive odd numbers beginning at 1 (eg 1, 3, 5, ..) always adds up to a perfect square

Let S=1+3+5+.......to n terms.
Common difference =2,so the nth term=1+(n-1)*2=2n-1
S=1+3+5+......+(2n-5)+(2n-3)+(2n-1)
writing in reverse order
S=(2n-1)+(2n-3)+(2n-5)+.......5+3+1
Adding the two series
2S=2n+2n+2n+..........to n terms=2n.2
Therefore S=n^2

Posted by danish ahmed khan on 2012-10-23 14:38:05