Integers with a bounded square root

Statement

For how many positive integers $n$ is $7 < \sqrt{n} < 20$?

Solution

Since all quantities are positive, we may square the chain without changing the directions: $7 < \sqrt{n} < 20 \iff 49 < n < 400.$ The integers satisfying $49 < n < 400$ (endpoints excluded) run from $50$ to $399$. Their count is $399 - 50 + 1 = 350.$ Check: $\sqrt{49} = 7$ and $\sqrt{400} = 20$ are excluded, so the first usable integer is $50$ and the last is $399$. Correct. $\boxed{350 \text{ integers}}$