<s>[INST] Find the value of a succession of additions $$1^2+2^2+3^2+...+10000$$
How do you find the exact value of that?
I'm studying induction, and I'm still not sure how to get that value.
 [/INST] Let $$S=1+2+3+...+n$$ then, $$S=n+(n-1)+(n-2)+...+1$$ Adding corresponding terms of two equations gives , $$2S=(n+1)+(n+1)+(n+1)+...(n+1)=n(n+1)\implies S=n(n+1)/2$$ Now $$(n+1)^3-n^3=3n^2+3n+1$$ $$\implies 2^3-1^3=3(1)^2+3(1)+1$$ $$3^3-2^3=3(2)^2+3(2)+1$$ $$4^3-3^3=3(3)^2+3(3)+1$$ $$..$$$$..$$$$(n+1)^3-n^3=3(n)^2+3(n)+1$$ Adding corresponding terms gives, $$(n+1)^3-1^3=3(1^2+2^2+...+n^2)+3(1+2+...+n)+n$$ $$\implies n^3+3n^2+3n=3(ReqSum)+3n(n+1)/2+n$$ Solving for $ReqSum$ and factorization gives $ReqSum=n(n+1)(2n+1)/6$. For $n=100$, it is $100.101.201/6=338350$. 
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