### 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 $\sqrt{2}$?

# Staircase

##### Stage: 5 Challenge Level:

This problem was suggested by Yatir Halevi.

Solve the equation:

$$x^{(x^3)}=3.$$

Solve the equation:

$$x^{(x^{(x^3)})}=3.$$

Can you find solutions to all equations of the form:

$$x^{x^{x^{x^{x^{x^{...^{a}}}}}}}=a$$

where the sequence of powers is defined in the same way and $a$ is a positive integer?

Explain what you have done and prove that you have found all possible solutions.