How do I prove that this finite sum evaluate to $1+\sqrt{2}$ for all values of n?

Question

$$ \frac{\displaystyle\sum_{i=1}^{n-1} \sqrt{\sqrt{n} + \sqrt{i}}} {\displaystyle\sum_{i=1}^{n-1} \sqrt{\sqrt{n} - \sqrt{i}}} = 1 + \sqrt{2} $$

When $n=2$, this is easy to verify. As $n \to \infty$, we can turn this into a Riemann sum that gives the same limit. But seems this is true for all $n \geq 2$. How can this be proved?