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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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Growing

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

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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 $\sqrt{2}$?

Power Quady

Age 16 to 18 Challenge Level:
You may wish to take a look at the similar but extended problem Mega Quadratic Equations.

Why do this problem?
This is a short problem that can be used as a lesson starter and it is non-standard so the learners have to think how to apply what they know. The problem also requires the learner to work systematically to be sure they have considered all possible cases and found all the solutions.

Possible approach
Let the class work on the problem and then make a list of all the solutions that they have found.

Key Questions
Have we found all possible solutions?
How can we be sure?