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: 90°, (45+x)° and (45-x)° in a triangle?

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

Sum the series $$1 \times 1! + 2 \times 2! + 3 \times 3! +...+n \times n!$$