Can anyone please check/confirm my work on this proof?

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Math100
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Homework Statement
Prove 1^2+2^2+...+n^2=(1/6)n(n+1)(2n+1) for all positive integers n.
Relevant Equations
None.
Please check/confirm my work of this proof and tell me if it's correct or not. Thank you.
 

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Your proof is not correct. Your final equation is not equal to the formula for n+1, and you are trying to prove the wrong thing anyway. The series is 12 + 22 +...+ n2, not 1 + 2 +...+n.
 
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There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
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