### Telescoping Series

Find $S_r = 1^r + 2^r + 3^r + ... + n^r$ where r is any fixed positive integer in terms of $S_1, S_2, ... S_{r-1}$.

### Degree Ceremony

Can you find the sum of the squared sine values?

### OK! Now Prove It

Make a conjecture about the sum of the squares of the odd positive integers. Can you prove it?

# The Root of the Problem

##### Age 14 to 18Challenge Level

The Root of the Problem printable sheet

Alison has been exploring sums with surds. She used a spreadsheet to make columns for square roots, and then added together various combinations.

Here is one of the sums she worked out: $$\frac{1}{\sqrt{1}+ \sqrt{2}}+ \frac{1}{\sqrt{2}+ \sqrt{3}} + ... +\frac{1}{ \sqrt {99}+ \sqrt{100}}.$$

Knowing that $(a+b)(a-b)=a^2-b^2$ might help.