Sum of Sequence of Seventh Powers

Theorem

 * $\displaystyle \sum_{j \mathop = 0}^n j^7 = \dfrac {n^2 \paren {n + 1}^2 \paren {3 n^4 + 6 n^3 - n^2 - 4 n + 2} } {24}$

Proof
The proof proceeds by induction.

For all $n \in \Z_{\ge 0}$, let $\map P n$ be the proposition:
 * $\displaystyle \sum_{j \mathop = 0}^n j^7 = \dfrac {n^2 \paren {n + 1}^2 \paren {3 n^4 + 6 n^3 - n^2 - 4 n + 2} } {24}$

$\map P 0$ is the case:

Thus $\map P 0$ is seen to hold.

Basis for the Induction
$\map P 1$ is the case:

Thus $\map P 1$ is seen to hold.

This is the basis for the induction.

Induction Hypothesis
Now it needs to be shown that, if $\map P k$ is true, where $k \ge 1$, then it logically follows that $\map P {k + 1}$ is true.

So this is the induction hypothesis:
 * $\displaystyle \sum_{j \mathop = 0}^k j^7 = \dfrac {k^2 \paren {k + 1}^2 \paren {3 k^4 + 6 k^3 - k^2 - 4 k + 2} } {24}$

from which it is to be shown that:
 * $\displaystyle \sum_{j \mathop = 0}^{k + 1} j^7 = \dfrac {\paren {k + 1}^2 \paren{k + 2}^2 \paren {3 \paren {k + 1}^4 + 6 \paren {k + 1}^3 - \paren {k + 1}^2 - 4 \paren {k + 1} + 2} } {24}$

Induction Step
This is the induction step:

So $\map P k \implies \map P {k + 1}$ and the result follows by the Principle of Mathematical Induction.

Therefore:
 * $\forall n \in \Z_{\ge 0}: \displaystyle \sum_{j \mathop = 0}^n j^7 = \dfrac {n^2 \paren {n + 1}^2 \paren {3 n^4 + 6 n^3 - n^2 - 4 n + 2} } {24}$