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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
You may wish to start by exploring using a spreadsheet.
Think about other ways to write $\frac{1}{\sqrt{n}+\sqrt{n+1}}$, particularly ways that have no surds in the denominator.