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My professor gave us a puzzle problem that we discussed in class that I could elaborate on if requested. But I interpreted the puzzle and formed a recursive function to model it which is as follows: $$f(n) = \frac{n f(n-1)}{n - 1} + .01 \hspace{1cm} \textrm{where } f(1) = .01\text{ and } n\in\mathbb{N}.$$

The question that is asked is when (if ever) does $f(x) = 1000x$. About half of the students concluded that it eventually will equal (they didn't have the formula I made) and that x would be near infinity.

My personal question is, can the function be reduced so that it isn't recursive? And so that it doesn't need to be solved by brute force computer algorithm (which would be about 3 lines of code).

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    Well, certainly we do get $f(x)\geq 1000x$ at some point, since at every stage we are multiplying the previous value by a number greater than $1$, and then adding more.2011-02-04
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    The backstory of the problem is this: an ant is travelling along a string a 1 cm/sec and the string is extended by 1 kilometer at the end of every second. Assume uniform stretching and no general relativity, etc. Does he reach the end of the string?2011-02-04

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