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This problem follows on from Power Mad!

Work out $9^n + 1^n$ for a few odd values of $n$.
What do you notice?
Can you prove it?

Now try the same with the following:
$7^n + 3^n$, where $n$ is odd.
$8^n - 2^n$, where $n$ is even.
$6^n - 4^n$, where $n$ is even.

Can you find any more results like these?