<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] The sum of squares formula is
$\sum_{i= 1}^n i^2 = \frac{n(n + 1)(2n + 1)}{6}$
Clearly it holds for $n = 1$. 
Suppose it held for $n$. That is
$\sum_{i= 1}^n i^2 = \frac{n(n + 1)(2n + 1)}{6}$
Then $\sum_{i = 1}^{n + 1} i^2 = \sum_{i = 1}^n i^2 + (n + 1)^2 = \frac{n(n + 1)(2n + 1)}{6} + (n + 1)^2 = \frac{2n^3 + 2n + n}{6} + \frac{6(n^2 + 2n + 1)}{6}$
$=\frac{2n^3 + 9n^2 + 13n + 6}{6} = \frac{(n + 1)((n + 1) + 1)(2(n + 1) + 1)}{6}$
Thus by induction the formula holds. 
Your series above is $\sum_{i = 1}^{100} i^2 = \frac{(100)(101)(201)}{6}$
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