# Math Help - a proof

1. ## a proof

Use mathematical induction to prove that 1^2 + 2^2 + 3^2.... + n^2 = (n(n+1)(2n+1))/6

thanks!

2. Originally Posted by s0urgrapes
Use mathematical induction to prove that 1^2, 2^2, 3^2....n^2 = (n(n+1)(2n+1))/6
It should be 1^2+ 2^2+ 3^2+....+n^2 = (n(n+1)(2n+1))/6

3. It's obviously true for n=1.

Assume $1^{2}+2^{2}+3^{2}+k^{2} = \frac {k(k+1)(2k+1)}{6}$ is true for some k>1.

then $1^{2}+2^{2}+3^{2}+k^{2}+(k+1)^{2} = \frac {k(k+1)(2k+1)}{6} + (k+1)^{2}$

$= \frac{2k^{3}+9k^{2}+13k+6}{6}$

$= \frac{(k+1)(2k^{2}+7k+6)}{6}$

$= \frac {(k+1)(k+2)(2k+3)}{6}$

$= \frac{\big(k+1\big)\big((k+1)+1\big)\big(2(k+1)+1\ big)}{6}$