Telescoping Series

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

Growing

Which is larger: (a) 1.000001^{1000000} or 2? (b) 100^{300} or 300! (i.e.factorial 300)

$2\wedge 3\wedge 4$ could be $(2^3)^4$ or $2^{(3^4)}$. Does it make any difference? For both definitions, which is bigger: $r\wedge r\wedge r\wedge r\dots$ where the powers of $r$ go on for ever, or $(r^r)^r$, where $r$ is $\sqrt{2}$?

Power Quady

Age 16 to 18 Challenge Level:

Find all real solutions of the equation
$$(x^2-7x+11)^{(x^2-11x+30)} = 1.$$

You may also be interested in the problem Mega Quadratic Equations.

Creating and manipulating expressions and formulae. Powers & roots. Number theory. Mathematical reasoning & proof. Golden ratio. Indices. Factors and multiples. Modular arithmetic. Quadratic equations. Inequalities.

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Telescoping Series

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Power Quady

Age 16 to 18 Challenge Level: