Ok, I've got another one...?

Question

why does

1+3+5+7+11+...+(2k-1)
=k^2 ?

Answer 1

Well you have:
(x=1, k)∑2x-1
By linearity of summation:
2*(x=1, k)∑x - (x=1, k)∑1
Simplifying:
2*(k(k+1))/2 - k
k(k+1) - k
k²+k-k
k²

Answer 2

1+3+5+...+(2k-1) = Sum(i=1, i=k, (2k-1))
1+3+5+...+(2k-1) = Sum(i=1, i=k, 2k) - Sum(i=1, i=k, 1)
1+3+5+...+(2k-1) = 2*Sum(i=1, i=k, k) - Sum(i=1, i=k, 1)
1+3+5+...+(2k-1) = 2*k*(k+1)/2 - k
1+3+5+...+(2k-1) = k^2 + k - k
1+3+5+...+(2k-1) = k^2