Consider the sum of n odd numbers
We can pair up symmetrical elements
that add to a certain constant and
then multiply by the number of pairs
(ie Gauss)
The largest odd number is 2n - 1. If
n is even, the sum S:
S = each pair * number of pairs
S = (2n-1 + 1) * (n / 2)
S = 2n * n/2 = nē
If n is odd, the numbers do not pair
out exactly, there is a remaining
central term. However note that this
term occurs in the middle so it is
equal to 2n/2 = n.
S = (each pair * number of pairs) + center
S = 2n * (n-1/2) + n (remember n is odd)
S = 2n * (n-1) + n
S = nē - n + n = nē
For any n, the sum is always a perfect square.
(See the comments for other interesting ways to prove
this) |