Telescoping Square Roots Galore

This was tricky, because it's hard to see how the denominator changed so drastically. One would normally expect that the denominator was just $\sqrt{n} \times \sqrt{n+1}$ , but this is not the case.

First, we combine fractions to get

$\frac{ \sqrt{n+1} - \sqrt{n} } { \sqrt{n} \times \sqrt{n+1} }.$

Notice that this is not any of the options. We rationalize the numerator, by multiplying it with $\sqrt{ n+1} + \sqrt{n}$ , which gives us

$\frac{ \sqrt{n+1} - \sqrt{n} } { \sqrt{n} \times \sqrt{n+1} } \times \frac { \sqrt{ n+1} + \sqrt{n} } { \sqrt{ n+1} + \sqrt{n} } = \frac{ 1 } { \sqrt{n} \times \sqrt{n+1} \times ( \sqrt{ n+1} + \sqrt{n} ) }.$

Finally, we expand and multiply, to obtain

$\frac{1} { \sqrt{ n} \times (n+1) + \sqrt{n+1} \times n } .$

just do the normal subtraction and then rationalize the numerator.