Intermediate Mathematical Challenge 2017 Q24

The $n$th term in a certain sequence is calculated by multiplying together all the numbers $\sqrt{1+\frac{1}{k}}$, where $k$ takes all the integer values from $2$ to $n+1$ inclusive. For example, the third term in the sequence is $\sqrt{1+\frac{1}{2}}\times\sqrt{1+\frac{1}{3}}\times\sqrt{1+\frac{1}{4}}$.

Which is the smallest value of $n$ for which the $n$th term of the sequence is an integer?