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Summation of Powers

Here are two scanned images showing how I derived the summation formulas for n^2 and n^3

squares-s

cubes-s

Here is another way to find the formula for the summation of n^2:

We start by looking at the term function for the every other series:
t=\frac{n^2+n}{2}
and the term function for the summation of the squares:
t=n^2
They both have the term n^2
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With the following manipulation both term functions are equivalent
(\frac{n^2+n}{2}) (2)-n=n^2

Now to carry those operations over to the summation formula for the every other series to derive the formula for the summation of the squares.

We multiply by the constant 2 and then subtract the summation formula for n as opposed to just n as we are now manipulating the summation formula and not the term formula.

(\frac{n^3+3n^2+2n}{6})(2)-\frac{n^2+n}{2}

which yields to us the summation formula for the squares

\frac{2n^3+3n^2+2n}{6}

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