text: why is the sum of the first n odd numbers the square of n?   1+3+5+7+9+...........+(2n-1)  where   n=1,2,3,4,....  in ap first term a=1 and diff d=2  as we know in ap  (d-a)+(2d-a)+.........+(nd-a) if d>a and n=1,2,3,.. sum= n(n+1)d/2 -na    where  n=1,2,3..  substituting a=1 and d=2 ( for odd numbers) sum=n^2+n-n sum=n^2.