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.

Useless Numerology for 2016: Part 2

Published by John Quintanilla

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Published by John Quintanilla

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