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