### 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?

# Summing Geometric Progressions

##### Age 14 to 18 Challenge Level:
Watch the video below to see how Alison works out the sum of the first twenty terms of the sequence: $$2, 8, 32, 128, 512 ...$$

Can you adapt Alison's method to sum the following sequences?
• $3, 9, 27, 81, 243 ...$ up to the 15th term

• $5, 10, 20, 40, 80 ...$ up to the 12th term

• $\sum_{i=1}^{20}(3 \times 2^{i-1})$

• $\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16} ...$ up to the 10th term

Can you find an expression for the following sum up to the nth term? $$a + ar + ar^2 + ar^3 + ...$$