How can I show that these are equal?

Question

How can I show that:$$\sum_{k=1}^{n}({k^2})$$ Is equal to: $$\frac{n(n+1)(2n+1)}{6}$$ I know that I would apply the sum formula, should I also be using this formula? $$\sum_{k=1}^{n}k=\frac{n(n+1)}2$$

Answer 1

Use proof by induction

1 - Demonstrate that it is true for $n=1$

2 - Demonstrate that if it is true for $n=k$ it must also be true for $n=k+1$

This comes down to demonstrating ... $$ (k+1)^2+ \frac{k(k+1)(2k+1)}{6} \\= \frac{(k+1)(k+2)(2(k+1)+1)}{6}$$