The sum of 1+3+5+... is represented by ∑2n-1 where n runs from 1 to n. So the question is does this sum equal n², or
∑2n-1 = n² ?
Taking the 2 and the 1 out of the summation gives:
2(∑n)-n = n²
Moving the 2 and the -n to the other side gives:
∑n = (n²-n)/2 = n(n+1)/2