Here you need to start with

\begin{matrix} a (n) & = & 1 + 2 + 3 + . . . + n & = & \frac{1}{2} n (n + 1) \\ b (n) & = & 1^{2} + 2^{2} + 3^{2} + . . . + n^{2} & = & \frac{1}{6} n (n + 1) (2 n + 1) \\ c (n) & = & 1^{3} + 2^{3} + 3^{3} + . . . + n^{3} & = & \frac{1}{4} n^{2} (n + 1)^{2} . \end{matrix}

These are well known results found in many text books and you will find proofs in some problems on this website. See Picture Story and Natural Sum.