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

What does Pythagoras' Theorem tell you about these angles: 90Â°, (45+x)Â° and (45-x)Â° in a triangle?

Make a conjecture about the sum of the squares of the odd positive integers. Can you prove it?

Challenge Level

Prove that the sum of the reciprocals of the first $n$ triangular numbers gets closer and closer to $2$ as $n$ grows.

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NRICH is part of the family of activities in the Millennium Mathematics Project.

NRICH is part of the family of activities in the Millennium Mathematics Project.