Sum of Square Root Terms in AP

Rationalizing each term: $\frac{\sqrt{a_2} - \sqrt{a_1}}{d} + \frac{\sqrt{a_3} - \sqrt{a_2}}{d} + \dots + \frac{\sqrt{a_{n+1}} - \sqrt{a_n}}{d} = \frac{\sqrt{a_{n+1}} - \sqrt{a_1}}{d}$. Since $a_{n+1} = a_1 + nd \implies d = \frac{a_{n+1} - a_1}{n}$. Sum $= \frac{\sqrt{a_{n+1}} - \sqrt{a_1}}{\frac{a_{n+1} - a_1}{n}} = \frac{n(\sqrt{a_{n+1}} - \sqrt{a_1})}{(\sqrt{a_{n+1}} - \sqrt{a_1})(\sqrt{a_{n+1}} + \sqrt{a_1})} = \frac{n}{\sqrt{a_1} + \sqrt{a_{n+1}}}$.