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How do I come up with a function to count a pyramid of apples?
Proof that $\sum\limits_{k=1}^nk^2 = \frac{n(n+1)(2n+1)}{6}$?
Finite Sum of Power?

I know that the sum of the squares of the first n natural numbers is $\frac{n(n + 1)(2n + 1)}{6}$. I know how to prove it inductively. But how, presuming I have no idea about this formula, should I determine it? The sequence $a(n)=1^2+2^2+...+n^2$ is neither geometric nor arithmetic. The difference between the consecutive terms is 4, 9, 16 and so on, which doesn't help. Could someone please help me and explain how should I get to the well known formula assuming I didn't know it and was on some desert island?

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    The first chapter of *Concrete Mathematics* by Graham, Knuth, and Patashnik presents about seven different techniques for deriving this identity, so you might be interested to look at that.2012-08-16
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    Coincidentally, I uploaded a web page recently regarding this topic. There I don't prove the formula (or even present it), but I provide the student a geometric strategy for finding it. http://mathuprising.comlu.com/sumofsquares2014-08-19
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    You can find a formal approach for even higher powers [here](http://insight-things.com/sum-squares-cubes-higher-powers) as well as a nice geometric approach [here](http://insight-things.com/simply-explained-sum-of-squares).2016-03-10

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