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.
Not surprisingly, there’s a natural reason why these two expressions are equal. (However, there isn’t a natural reason why the answer happens to match the current year other than coincidence.)
To begin, is a finite geometric series. The first term is , the common ratio is , and there are terms in the series. Using the formula for a finite geometric series,
thus establishing that these two expressions are equal.