Question

The value of {1/(2+√7)}+{1/(√7+√10}+...........{1/(√28+√31)} is

Answer 1

we have to find the value of
$\frac{1}{\sqrt{4}+\sqrt{7}}+\frac{1}{\sqrt{7}+\sqrt{10}}+\frac{1}{\sqrt{10}+\sqrt{13}}+......\frac{1}{\sqrt{28}+\sqrt{31}}$

do rationalisation of each term,

$\frac{1}{\sqrt{4}+\sqrt{7}}=\frac{\sqrt{7}-\sqrt{4}}{3}$

$\frac{1}{\sqrt{7}+\sqrt{10}}=\frac{\sqrt{10}-\sqrt{7}}{3}$


$\frac{1}{\sqrt{10}+\sqrt{13}}=\frac{\sqrt{13}-\sqrt{10}}{3}$

.......... ....

$\frac{1}{\sqrt{28}+\sqrt{31}}=\frac{\sqrt{31}-\sqrt{28}}{3}$

adding all equations ,

$\frac{\sqrt{7}-\sqrt{4}+\sqrt{10}-\sqrt{7}+\sqrt{13}-\sqrt{10}+......+\sqrt{31}-\sqrt{28}}{3}$

$=\frac{\sqrt{31}-\sqrt{4}}{3}$

hence, (√31 - 2)/3 is the answer