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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 $S_1, S_2, ... S_{r-1}$.
Climbing Powers
$2\wedge 3\wedge 4$ 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 $\sqrt{2}$?
Age 16 to 18
Challenge Level
This problem was suggested by Yatir Halevi.
Solve the equation:
$$x^{(x^3)}=3.$$
Solve the equation:
$$x^{(x^{(x^3)})}=3.$$
Can you find solutions to all equations of the form:
$$x^{x^{x^{x^{x^{x^{...^{a}}}}}}}=a$$
where the sequence of powers is defined in the same way and $a$ is a positive integer?
Explain what you have done and prove that you have found all possible solutions.
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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.