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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 $S_1, S_2, ... S_{r-1}$.
Climbing Powers
$2\wedge 3\wedge 4$ 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 $\sqrt{2}$?
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?
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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NRICH is part of the family of activities in the Millennium Mathematics Project.
NRICH is part of the family of activities in the Millennium Mathematics Project.