Find the number of positive integers n such that $\sqrt{n+\sqrt{n+\sqrt{n}}}<10$ for any finite number of square root signs. Find the number of positive integers $n$ such that $\sqrt{n+\sqrt{n+\sqrt{n}}}<10$ for any finite number of square root signs.
I know something with squaring and repeating, but what does "finite number of square root signs" mean?

As the number of square roots goes from $1$ to $\infty$, the nested radical will increase monotonically from $\sqrt n$ to a limit of $\frac{1+\sqrt{1+4n}}2$. This means

*

*if a choice of $n$ produces a limit greater than $10$, it is inadmissible since some finite number of square roots will also produce a result greater than $10$

*if a choice of $n$ produces a limit less than $10$, it is admissible; all finite nested radicals of its form will produce lesser results

Now
$$\frac{1+\sqrt{1+4n}}2<10\implies n<90$$
This is the tighter bound on $n$, since $\sqrt n<10\implies n<100$. $n=90$ is also admissible though since we deal with a finite number of square roots, not the limit. There are thus $90$ admissible integers.
