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# Discrete Trends

##### Age 16 to 18 Challenge Level:

As $n$ is an integer try finding $n^{1\over n}$ for some small values of $n$. What do you find? If you think you might have found the maximum value then you'll need to use the first part of the question to prove it really is the maximum. As the problem is about discrete (whole number) values you can find a solution without calculus. To show that

$$n^{1/n} < 1 + \sqrt {{2\over {n-1}}}$$

write $n^{1/n} = 1 + \delta$ and use the Binomial Theorem. If $n> 1$ then $\delta> 0$. Throw away all but one term of the Binomial expansion to get the inequality.