Prove that $1^2 + 2^2 + ..... + (n-1)^2 < \frac {n^3} { 3} < 1^2 + 2^2 + ...... + n^2$ I'm having trouble on starting this induction problem.
The question simply reads : prove the following using induction: 
$$1^{2} + 2^{2} + ...... + (n-1)^{2} < \frac{n^3}{3} < 1^{2} + 2^{2} + ...... + n^{2}$$

If you wish a more direct application of induction, we have that if
$$
1^2+2^2+\cdots+(n-1)^2 < \frac{n^3}{3}
$$
then
$$
1^2+2^2+\cdots+n^2 < \frac{n^3+3n^2}{3}
                   < \frac{n^3+3n^2+3n+1}{3} = \frac{(n+1)^3}{3}
$$
Similarly, if
$$
1^2+2^2+\cdots+n^2 > \frac{n^3}{3}
$$
then
$$
1^2+2^2+\cdots+(n+1)^2 > \frac{n^3+3n^2+6n+3}{3}
                       > \frac{n^3+3n^2+3n+1}{3} = \frac{(n+1)^3}{3}
$$
