I am just learning about summations and their formulas. 
I would like to know the proof behind this formula in this example. It doesn't matter to me whether it is an intuitive one or a rigorous one. Thanks for all the help!
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3Induction is a reliable method for attacking things like this. – lulu Sep 12 '22 at 16:31
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Almost all such proofs are by induction. The real trick is figuring out what the right hand side should be, but that isn't necessary in a "prove this" question. – Thomas Andrews Sep 12 '22 at 16:38
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As the comment of @ThomasAndrews indicated, you have two choices for how to use induction on this problem: [1] Verify that $$\sum_{k=1}^n k^2 = \frac{1}{3}n^3 + \frac{1}{2}n^2 + \frac{1}{6}n$$ or [2] Derive that if $$\sum_{k=1}^n k^2 = c_0n^3 + c_1n^2 + c_2n,$$ then the only satisfying ordered triple $$(c_0,c_1,c_2) = \left(\frac{1}{3}, \frac{1}{2}, \frac{1}{6}\right).$$ – user2661923 Sep 12 '22 at 17:37