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**1. The problem statement, all variables and given/known data**

For a positive integer ##n##, let

##a_n=\frac{1}{n} \sqrt[3]{n^{3}+n^{2}-n-1}##

Find the smallest positive integer ##k \geq2## such that ##a_2a_3\cdots a_k>4##

**2. Relevant equations**

The restrictions are the only relevant thing I can think of

**3. The attempt at a solution**

I have just tried plugging in numbers so far

When ##n=2## I got ##\frac{3^{\frac{2}{3}}}{2}##

When ##n=3## I got ##\frac{2 \times 2^{\frac{2}{3}}}{3}##

When ##n=4## I got ##\frac{1}{4} 3^{\frac{1}{3}} \hspace{1mm} 5^{\frac{2}{3}}##

Now this is growing really slowly so this is obviously not the correct approach.