The standard method is induction and you should look it up as it is a popular second example (first is $\sum n$)

Another argument is use: $$24n^2 +2= (2n+1)^3-(2n-1)^3$$ and get a [telescoping][1] sum.

i.e $$24\sum_1^n k^2 +2n = \sum_1^n (2k+1)^3-\sum_1^n (2k-1)^3$$ $$24\sum_1^n k^2 +2n = (2n+1)^3-1$$ $$24\sum_1^n k^2 =8 n^3+12 n^2+4 n$$ $$24\sum_1^n k^2 =4 n (n+1) (2 n+1)$$ $$\sum_1^n k^2 = \frac{n (n+1) (2 n+1)}{6}$$ [1]: http://en.wikipedia.org/wiki/Telescoping_series

The standard method is induction and you should look it up as it is a popular second example (first is $\sum n$)

Another argument is use: $$24n^2 +2= (2n+1)^3-(2n-1)^3$$ and get a telescoping sum.

i.e $$24\sum_1^n k^2 +2n = \sum_1^n (2k+1)^3-\sum_1^n (2k-1)^3$$ $$24\sum_1^n k^2 +2n = (2n+1)^3-1$$ $$24\sum_1^n k^2 =8 n^3+12 n^2+4 n$$ $$24\sum_1^n k^2 =4 n (n+1) (2 n+1)$$ $$\sum_1^n k^2 = \frac{n (n+1) (2 n+1)}{6}$$

The standard method is induction and you should look it up as it is a popular second example (first is $\sum n$)

Another argument is use: $$24n^2 +2= (2n+1)^3-(2n-1)^3$$ and get a telescoping sum.

The standard method is induction and you should look it up as it is a popular second example (first is $\sum n$)

Another argument is use: $$24n^2 +2= (2n+1)^3-(2n-1)^3$$ and get a [telescoping][1] sum.

i.e $$24\sum_1^n k^2 +2n = \sum_1^n (2k+1)^3-\sum_1^n (2k-1)^3$$ $$24\sum_1^n k^2 +2n = (2n+1)^3-1$$ $$24\sum_1^n k^2 =8 n^3+12 n^2+4 n$$ $$24\sum_1^n k^2 =4 n (n+1) (2 n+1)$$ $$\sum_1^n k^2 = \frac{n (n+1) (2 n+1)}{6}$$ [1]: http://en.wikipedia.org/wiki/Telescoping_series