Suppose $n\geq 9$ is an integer. Let $\mu=n^{\frac{1}{2}}+ n^{\frac{1}{3}}+n^{\frac{1}{4}}$. Then which of the following relationships between $n$ and $μ$ is correct?
1)$n=\mu$
2)$n<\mu$
3)$n>\mu$
Relationship between $n\geq 9$ & $\mu=n^{\frac{1}{2}}+ n^{\frac{1}{3}}+n^{\frac{1}{4}}$
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problem-solving
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1As $n$ gets to be large \mu approaches $\sqrt n$. $\mu (9) < 9$ and $\mu$ is growing more slowly than $n.$ – 2017-02-23
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1Hint: find $\mu/n$ as $n\to\infty$. – 2017-02-23
1 Answers
2
Firstly we can rule out option 1 since if we take $n = 9$ then $\mu = 3 + 9^\frac{1}{3} + 9^\frac{1}{4} < 9$
This also rules out option 2 by counterexample. Now we are left with option 3.
Now note that $n^\frac{1}{3}, n^\frac{1}{4} < n^\frac{1}{2}$
Therefore we can consider $n - 3*n^\frac{1}{2}$
For $n \geqslant 9$:
$3*n^\frac{1}{2} $\Rightarrow n > n^\frac{1}{2} + n^\frac{1}{3} + n^\frac{1}{4} = \mu$ Therefore option 3 is correct.