Useless Numerology for 2016: Part 4

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 = 3^3 + 4^3 + 5^3 + 6^3 + 7^3 + 8^3 + 9^3

2016 = (1+2+...+8+9)^2 - (1+2)^2

To evaluate the top series, we use the formula

1^3 + 2^3 + \dots + n^3 = \displaystyle \frac{n^2 (n+1)^2}{4}

= \displaystyle \left( \frac{n(n+1)}{2} \right)^2

= (1 + 2 + \dots + n)^2,

where we used the formula for an arithmetic series. Therefore,

3^3 + 4^3 + 5^3 + 6^3 + 7^3 + 8^3 + 9^3 = (1^3 + 2^3 + 3^3 + 4^3 + 5^3 + 6^3 + 7^3 + 8^3 + 9^3) - (1^3 + 2^3)

= (1 + 2 + \dots + 9)^2 - (1 + 2)^2.

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