### Telescoping Series

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

### Growing

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

### 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 18Challenge Level

Find all real solutions of the equation
$$(x^2-5x+5)^{(x^2-11x+30)} = 1$$

There are six possible solutions to the equation - did you find all six?

Here are some more questions to think about

1. Find all the solutions to $(x^2 - 7x + 11)^{(x^2 - 13x + 42)} = 1$.
How do these solutions compare to the first equation?

2. Can you find a Mega Quadratic Equation with solutions $3, 4, 5, 6, 7, 8$?
How about $4, 5, 6, 7, 8, 9$?...

3. Can you explain why there are only $4$ solutions to $(x^2-5x+5)^{(x^2-4)}=1$?

4. Can you explain why there are only $3$ solutions to $(x^2-6x+10)^{(x^2+x-2)}=1$?

5. Can you find a Mega Quadratic equation with exactly $2$ solutions? $5$ solutions?

With thanks to Don Steward, whose ideas formed the basis of this problem.