<s>[INST] How do I prove by induction? For example if i wanted to prove: 
$1^2 + \dots + n^2 = \frac {n(n + 1)(2n + 1)} {6}$
by induction.
I'm not sure where to start. 
Thanks. 
 [/INST] Since $\frac{1(1 +1)(2+1)}{6} = 1$, then base case holds. Suppose result holds for $n$. Then
$$\sum_{i=1}^{n+1} i^2 = \sum_{i=1}^{n}i^2 + (n+1)^2 = \frac{n(n+1)(2n + 1)}{6} + (n+1)^2 $$
$$ \frac{n(n+1)(2n + 1)}{6} + (n+1)^2 = \frac{(n+1)[n(2n+1) + 6(n+1)}{6} = \frac{(n+1)(2n^2 + 7n + 1)}{6} =_{*} \frac{(n+1)(n+2)(2n + 3)}{6} = \frac{(n+1)((n+1) + 1)(2(n+1) + 1)}{6} $$
(*): Since $2n^2 + 7n + 1 = (n+2)(2n+3)$. The problem is solved by math induction.
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