### 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: 90Â°, (45+x)Â° and (45-x)Â° 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?

# Summing Geometric Progressions

##### Age 14 to 18Challenge Level
Aswaath, from Garden International School, Kuala Lumpur, adapted Alison's method and then used a spreadsheet to calculate the sums directly to verify that Alison's method worked.

Well done to Matthew from Westlakes Academy, Andrew from West Island School, Hong Kong, Michael, and Irina from Loughborough High School for their correct solutions. You can read Michael's solution and Irina's solution.