### 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 $\sqrt{2}$?

# Tens

##### Stage: 5 Challenge Level:

Why do this problem?
There are eight results to be proved. Learners are invited to spot patterns, to make conjectures and then to prove their conjectures.

The two principal methods of proof involve ideas central to mathematics courses in the last two years of school. Finding both methods is one of the challenges here.

Quite apart from the two key methods of proof there is some algebraic manipulation to do involving the laws of indices. As the algebra involves different powers of whole numbers it is relatively straightforward and so the problem provides good practice in working with indices for students who need such practice.

Possible approach
If students are proficient they won't need to prove all 8 cases once they have demonstrated that they can prove one, but if they do not find the proofs easy then it will be good practice to prove the results in all the cases.

If none of the class uses the Binomial Theorem the teacher may like to suggest this as an alternative method giving practice with the Binomial Theorem and a neat and short solution.

Key question
Can we use the fact that the numbers occurring in each part add up to 10?