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Mega Quadratic Equations
Age 14 to 18
Challenge Level 
Do you know any values of $a$ and $b$ that satisfy $a^b=1$?
If $a=1$, what could $b$ be?
If $a=-1$, what could $b$ be?
If $a=2$ what must $b$ be?
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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 $S_1, S_2, ... S_{r-1}$.
