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Here you need to start with $$\begin{eqnarray} a(n) &= &1 + 2 + 3 + ... + n &= & {1\over 2}n(n+1)\\ b(n) &= &1^2 + 2^2 + 3^2 + ... + n^2 &= & {1\over 6}n(n+1)(2n + 1)\\ c(n) &= &1^3 + 2^3 + 3^3 + ... + n^3 &= & {1\over 4}n^2(n+1)^2. \end{eqnarray}$$
More Polynomial Equations
Age 16 to 18
Challenge Level 
Here you need to start with $$\begin{eqnarray} a(n) &= &1 + 2 + 3 + ... + n &= & {1\over 2}n(n+1)\\ b(n) &= &1^2 + 2^2 + 3^2 + ... + n^2 &= & {1\over 6}n(n+1)(2n + 1)\\ c(n) &= &1^3 + 2^3 + 3^3 + ... + n^3 &= & {1\over 4}n^2(n+1)^2. \end{eqnarray}$$
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.
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