### 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 $\sqrt{2}$?

# Giants

##### Stage: 5 Challenge Level:

Why do this problem?

Learners will no doubt happily do the first part using a calculator but for the other two parts the numbers are too big for the calculator to cope with directly and learners really have to think and to apply what they know about indices.

It is also instructive to compare different methods of solution. One method relies on understanding place value and the way in which raising a number to a given power involves multiplication. Another method involves a manipulation of the inequality relation and a good understanding of indices and simple inequalities.

School mathematics should give learners plenty of experience of inequalities as they are of central importance in mathematics beyond school.

Possible approach

This problem needs no introduction from the teacher. It provides a good lesson starter. Discussion of this problem can be used by way of review of the learners' prior knowledge of indices before the teacher introduces the topic to the class.

Key questions

What does $99^{100}$ mean?
How many digits does it have?
What happens when you raise $100^{1/100}$ to the power 9900?

Extension

For learners who know about logarithms pose the extension challenge for a string of a billion 9s and a string of a billion-and-1 10s