Question

Let $a_1, a_2, \ldots, a_n$ be in A.P. If $a_5=2 a_7$ and $a_{11}=18$, then $12\left(\frac{1}{\sqrt{a_{10}}+\sqrt{a_{11}}}+\frac{1}{\sqrt{a_{11}}+\sqrt{a_{12}}}+\ldots+\frac{1}{\sqrt{a_{17}}+\sqrt{a_{18}}}\right)$ is equal to

Answer 1

$2 a_5=a_5(\text { given }) $
$ 2\left(a_1+6 d\right)=a_1+4 d $
$ a_1+8 d=0 $....(1)
$ a_1+10 d=18 $...(2)
$ \text { By }(1) \text { and (2) we get } a_1=-72, d=9 $
$ a_{18}=a_1+17 d=-72+153=81 $
$ a_{10}=a_1+9 d=9 $
$ 12\left(\frac{\sqrt{a_{11}}-\sqrt{a_{10}}}{d}+\frac{\sqrt{a_{12}}-\sqrt{a_{11}}}{d}+\ldots \ldots \frac{\sqrt{a_{18}}-\sqrt{a_{17}}}{d}\right) $
$ 12\left(\frac{\sqrt{a_{18}}-\sqrt{a_{10}}}{d}\right)=\frac{12(9-3)}{9}=\frac{12 \times 6}{6}=8$