The sum of the first consecutive odd numbers is a square number.
Why? What do perfect squares have to do with odd numbers? At first glance, these are two seemingly unrelated types of numbers.
Some of us (okay, me) may have presented something like this:
1 + 3 + 5 + … + (2n – 1)
(2n – 1) + … + 5 + 3 + 1
The sum of each column is 2n. We have n columns. The total is then n × 2n = 2n². We added the sum twice so 2n² ÷ 2 = n².
Can you see what perfect squares have to do with odd numbers? Me neither.
Compare that with the following explanation¹ given in Paul Lockhart’s “A Mathematician’s Lament”.