Sum of first n odd numbers = n^2
Why is the sum of the first n odd numbers n^2?
@tidyboar This is a very cool fact! It relates to the fact that if you start with one square, \(n^2,\) and want to get the next consecutive square, \((n+1)^2,\) you just need to add \( n + (n+ 1).\)
\( n + n + 1 = 2n + 1\) is always odd, so to get the next square, we always add an odd number, and this odd number increases by \(2\) each time. To get the next square after that, we would add \( (n + 1) + (n + 2) = 2n + 3,\) which is \(2\) larger than the previous. There's a formal proof for why the sum of the first \(n\) odd integers equals \(n^2,\) using induction, but hopefully this brief explanation can shed some light on why this is true!
@debbie ooooh cool!