# Useless Numerology for 2016: Part 3

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.

$2016 = 1+2+3 + \dots + 62 + 63$

$2016 = 2^{11} - 2^5$

In this post, we’ll explore why these two expressions have to be equal.

The sum $1 + 2 + 3 + \dots + 62 + 63$ is an arithmetic series. The first term is $1$, the last term is $63$, and there are $63$ terms in the series. Using the formula for an arithmetic series, we find

$1 + 2 + 3 + \dots + 62 + 63 = \displaystyle \frac{(63)(1 + 63)}{2}$

$= \displaystyle \frac{63 \times 64}{2}$

$= 63 \times 32$

$= (64-1) \times 32$

$= (2^6 - 1) \times 2^5$

$= 2^{11} - 2^5$.