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

Degree Ceremony

What does Pythagoras' Theorem tell you about these angles: 90°, (45+x)° and (45-x)° in a triangle?

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 18 Challenge Level:

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: $$\frac{1}{\sqrt{1}+ \sqrt{2}}+ \frac{1}{\sqrt{2}+ \sqrt{3}} + ... +\frac{1}{ \sqrt {99}+ \sqrt{100}}.$$

The answer surprised her!
Can you find a way to evaluate the sum without using a calculator or spreadsheet?

Click here for a hint:

When a fraction contains surds, we often choose to multiply the numerator and denominator by an expression that gets rid of any surds in the denominator.
Knowing that $(a+b)(a-b)=a^2-b^2$ might help.

Can you find other similar sums with surds that give whole number answers?