Induction proof $\sum_{k=1}^{n} k^{2} = \frac{1}{6} n(n+1)(2n+1)$ I'm asked to prove $$\sum_{k=1}^{n} k^{2} = \frac{1}{6} n(n+1)(2n+1)$$ 
using proof by induction.
Now, I know how to do induction proofs and I end up at this step, needing to prove that: 
$$\frac{1}{6} n(n+1)(2n+1) + (n+1)^2 = \frac{1}{6} (n+1)(n+2)(2n+3)$$
So $$n(2n+1) + n+1 = (n+2)(2n+3)$$
$$2n^2 + 2n + 1 = 2n^2 + 7n + 6$$
which is obviously not equal. 
What am I doing wrong?

There is a simple visual proof of this: link

