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.

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