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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?

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?

OK! Now Prove It

Age 16 to 18
Challenge Level

Are there any standard summations of series you can use to help you?

Once you have a conjecture, proof by induction is a possibility. You can read more about the technique in this article.

See also Natural Sum, Picture Story and Polynomial Equations.