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# Tens

*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?

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Age 16 to 18

Challenge Level

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?

Find $S_r = 1^r + 2^r + 3^r + ... + n^r$ where r is any fixed positive integer in terms of $S_1, S_2, ... S_{r-1}$.

$2\wedge 3\wedge 4$ could be $(2^3)^4$ or $2^{(3^4)}$. Does it make any difference? For both definitions, which is bigger: $r\wedge r\wedge r\wedge r\dots$ where the powers of $r$ go on for ever, or $(r^r)^r$, where $r$ is $\sqrt{2}$?