The following entertaining (but useless) facts about the number 2,016 appeared in a recent Facebook post (and subsequent comments) by the American Mathematical Monthly.
Establishing that these two expressions are equal takes a little bit of work. We begin by dividing the sum into even and odd values of . The odd values of from to have the form , where varies between and . The even values of from to have the form , where again varies between and .
This last sum is an arithmetic series. The first () term is , the last term is , and there are terms. Using the formula for an arithmetic series, we find