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# OK! Now Prove It

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### Telescoping Series

### Overarch 2

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Age 16 to 18

Challenge Level

- Problem
- Getting Started
- Student Solutions

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.

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

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?