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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 $S_1, S_2, ... S_{r-1}$.


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

Remainder Hunt

What are the possible remainders when the 100-th power of an integer is divided by 125?


Age 16 to 18 Challenge Level:

Why do this problem?
The problem gives practice in using the notation for Binomial coefficients and manipulating algebraic expressions. In problem solving mode, if they can't get started, they might first try to work on the formula for small integer values of $n$.

Possible approach
Use as a revision exercise.

Key questions

If ${2n \choose n}$ is a binomial coefficient in the expansion of some power of $(1 + x)$ what can you say about the expansion and about the term where it occurs?

What do we know about ${n\choose r}$ and ${n\choose n-r}$?
Possible support

Ask learners to find the coefficient of $x^2$ in the expansion of $(1+x)^4$, the coefficient of $x^3$ in the expansion of $(1 +x)^6$ and then the coefficient of $x^4$ in the expansion of $(1 + x)^8$ and then ask them to try to connect their results to the problem given.

It might help to do the problem Summit first.

You could ask students to show that the sum of the $n$th row in Pascal's Triangle is $2^n$ first - so that they have a sense of achievement even if they don't succeed in proving the result in this problem.