Sum of Sequences of Fifth and Seventh Powers

Theorem

 * $\displaystyle \sum_{i \mathop = 1}^n i^5 + \sum_{i \mathop = 1}^n i^7 = 2 \left({\sum_{i \mathop = 1}^n i}\right)^4$

Proof
Proof by induction:

For all $n \in \N_{> 0}$, let $P \left({n}\right)$ be the proposition:
 * $\displaystyle \sum_{i \mathop = 1}^n i^5 + \sum_{i \mathop = 1}^n i^7 = 2 \left({\sum_{i \mathop = 1}^n i}\right)^4$

Basis for the Induction
So $P \left({1}\right)$ has been demonstrated to hold.

This is our basis for the induction.

Induction Hypothesis
Now we need to show that, if $P \left({k}\right)$ is true, where $k \ge 1$, then it logically follows that $P \left({k+1}\right)$ is true.

So this is our induction hypothesis:
 * $\displaystyle \sum_{i \mathop = 1}^k i^5 + \sum_{i \mathop = 1}^k i^7 = 2 \left({\sum_{i \mathop = 1}^k i}\right)^4$

Then we need to show:
 * $\displaystyle \sum_{i \mathop = 1}^{k+1} i^5 + \sum_{i \mathop = 1}^{k+1} i^7 = 2 \left({\sum_{i \mathop = 1}^{k+1} i}\right)^4$

Induction Step
This is our induction step:

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

Therefore:
 * $\displaystyle \forall n \in \N_{>0}: \sum_{i \mathop = 1}^n i^5 + \sum_{i \mathop = 1}^n i^7 = 2 \left({\sum_{i \mathop = 1}^n i}\right)^4$