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# Staircase

Consider this as a sequence given by:

$$x_{n+1}=x^{x_n}$$

where $x_1=x^3$.

Now consider the sequence of equations given by $x_n=3$.

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Age 16 to 18

Challenge Level

- Problem
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Consider this as a sequence given by:

$$x_{n+1}=x^{x_n}$$

where $x_1=x^3$.

Now consider the sequence of equations given by $x_n=3$.

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

$2\wedge 3\wedge 4$ 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 $\sqrt{2}$?