Useless Numerology for 2016: Part 2

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 = 2^{10} + 2^9 + 2^8 + 2^7 + 2^6 + 2^5

2016 = 2^{11} - 2^5

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, 2^5 + 2^6 + 2^7 + 2^8 + 2^9 + 2^{10} is a finite geometric series. The first term is 2^5 = 32, the common ratio is 2, and there are 6 terms in the series. Using the formula for a finite geometric series,

2^5 + 2^6 + 2^7 + 2^8 + 2^9 + 2^{10} = \displaystyle \frac{2^5 (1-2^6)}{1-2} = \frac{2^5-2^{11}}{-1} = 2^{11} - 2^5,

thus establishing that these two expressions are equal.

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