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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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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 $\sqrt{2}$?


Age 16 to 18 Challenge Level:

Why do this problem?

As well as offering practice in working with indices, the proofs in this problem can be done using several different techniques, using the binomial theorem or modular arithmetic.

Possible approach

See Power Mad! for a related problem that you may wish to set students instead of or as well as this one.

Start by inviting students to explore $9^n+1^n$ for odd values of $n$. When they observe that the answers all seem to be multiples of $10$, give students a short while to come up with their own convincing arguments. They are likely to use an argument based on the last digits of odd powers of $9$, and this could be a good opportunity to introduce the language and notation of modular arithmetic, and the idea of working mod $10$. This article could be given to students as background reading.

Alternatively, if you wish to focus on using the binomial theorem, invite students to express the questions in terms that can be expanded: for example, rewriting $7^n + 3^n$ as $7^n + (10-7)^n$. Students can write out the expression for odd and even values of $n$ and construct a convincing justification why the expression is always a multiple of $10$ when $n$ is odd.

Key questions

What happens to the units digit when you find consecutive powers of a number?
Can we use the fact that the numbers occurring in each part add up to 10?

Possible extension

For some challenging work on powers, see Giants.

Possible support

Power Mad! provides a slightly more playful introduction to the ideas met in this problem.