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Speedy Summations
Age 16 to 18
Challenge Level 
If you are finding it hard to get started, look at Slick Summing first.
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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?