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Power Quady

Find all real solutions of the equation (x^2-7x+11)^(x^2-11x+30) = 1.

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Telescoping Series

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

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Which is larger: (a) 1.000001^{1000000} or 2? (b) 100^{300} or 300! (i.e.factorial 300)


Age 16 to 18 Challenge Level:
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?