This looks like an easy question but I can't mathematically generalize it. I need to form a pattern and prove it. So far I have observed the pattern of [153, 165033, 166500333, 166650003333, ...], but I can't seem to mathematically come to a generalization
Prove a generalization of a sum of cubes forming a pattern
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sequences-and-series
algebra-precalculus
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0Flagged for moderator attention. Like the [other one](http://math.stackexchange.com/questions/2172935/are-there-any-repeats-of-a-number-eg-repeat-of-86-is-8686-that-are-perfect-squ) you posted, this question is part of the [PROMYS 2017](http://promys.org/program/applications) [application problem set](http://promys.org/sites/promys.org/files/assets/Problems2017.pdf) which is still ongoing with a submission deadline of April 1st 2017. At the risk of repeating my other comment: this site doesn't condone cheating. – 2017-03-08
1 Answers
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(Using the ``floor'' function) I think this is an expression for your pattern for a given positive integer $n$
$$\left\lfloor \frac{10^n}{6}\right\rfloor^3+\left\lfloor\frac{10^n}{2}\right\rfloor^3+\left\lfloor\frac{10^n}{3}\right\rfloor^3= \left\lfloor \frac{10^n}{6}\right\rfloor 10^{2n} + \left\lfloor \frac{10^n}{2}\right\rfloor 10^{n}+\left\lfloor \frac{10^n}{3}\right\rfloor.$$
Maple verified that this works for $n=1, 2, \ldots, 10$. I didn't try to prove it.