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

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
Draw a right-angled triangle with angles $90^\circ$, $(45+x)^\circ$ and $(45-x)^\circ$.

What does Pythagoras' Theorem tell you about these angles?

Use this information to find $ \sin^2 1^\circ + \sin^2 2^\circ + \, \cdots \,+ \sin^2 359 ^\circ + \sin^2 360^\circ$.