Question

The value of $\frac{1}{2+\sqrt{7}}+\frac{1}{\sqrt{7}+\sqrt{10}}+...+\frac{1}{\sqrt{28}+\sqrt{31}}$ is?

• A. $\frac{4-\sqrt{28}}{3}$
• B. $\frac{2-\sqrt{31}}{3}$
• C. $\frac{\sqrt{28}-4}{3}$
• D. $\frac{\sqrt{31}-2}{3}$

Answer 1

The correct option is D $\frac{\sqrt{31}-2}{3}$

$\frac{1}{2+\sqrt{7}}+\frac{1}{\sqrt{7}+\sqrt{10}}+...........+\frac{1}{\sqrt{28}+\sqrt{31}}$

$=(\frac{1}{2+\sqrt{7}}\times \frac{2-\sqrt{7}}{2-\sqrt{7}})+(\frac{1}{\sqrt{7}+\sqrt{10}}\times \frac{\sqrt{7}-\sqrt{10}}{\sqrt{7}-\sqrt{10}})+..........+(\frac{1}{\sqrt{28}+\sqrt{31}}\times \frac{\sqrt{28}-\sqrt{31}}{\sqrt{28}-\sqrt{31}})$

$=\frac{2-\sqrt{7}}{4-7}+\frac{\sqrt{7}-\sqrt{10}}{7-10}+........+\frac{\sqrt{28}-\sqrt{31}}{28-31}$

$=(-\frac{2}{3}+\frac{\sqrt{7}}{3})+(-\frac{\sqrt{7}}{3}+\frac{\sqrt{10}}{3})+........+(-\frac{\sqrt{28}}{3}+\frac{\sqrt{31}}{3})$

$=-\frac{2}{3}+\frac{\sqrt{31}}{3}$

$=\frac{\sqrt{31}-2}{3}$