### Telescoping Series

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

### OK! Now Prove It

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

### Overarch 2

Bricks are 20cm long and 10cm high. How high could an arch be built without mortar on a flat horizontal surface, to overhang by 1 metre? How big an overhang is it possible to make like this?

# Degree Ceremony

##### Age 16 to 18 Challenge Level:

You can draw the graph of the sine function. What about the graph of $y = \sin^2 x$? What about the periodicity of this graph?

What can you say about the angles (45 + x) degrees and (45 - x) degrees, the sines of these angles and the squares of the sines of these angles?