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### Number and algebra

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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

Notice that $$1^2 = {1\times 2\times 3 \over 6}$$ $$1^2 + 3^2 =
{3\times 4\times 5 \over 6}$$ $$1^2 + 3^2 + 5^2 = {5\times 6\times
7 \over 6}.$$ Make a conjecture about the sum $$1^2 + 3^2 + 5^2 +
\dots + (2n - 1)^2$$ and prove your conjecture.

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?