Question

Let $a_1, a_2, \ldots, a_n$ be in AP 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

Correct Answer: 8

The correct answer is 8.
$2a_5=a_5(\text{given})$
$2\left(a_1+6d\right)=a_1+4d$
$a_1+8d=0$....(1)
$a_1+10d=18$...(2)
$\text{By}(1)\text{and (2) we get}{a}_1=-72,d=9$
$a_{18}=a_1+17d=-72+153=81$
$a_{10}=a_1+9d=9$
$12\left(\frac{\sqrt{a_{11}}-\sqrt{a_{10}}}{d}+\frac{\sqrt{a_{12}}-\sqrt{a_{11}}}{d}+\dots \dots \frac{\sqrt{a_{18}}-\sqrt{a_{17}}}{d}\right)$
$12\left(\frac{\sqrt{a_{18}}-\sqrt{a_{10}}}{d}\right)=\frac{12(9-3)}{9}=\frac{12\times 6}{6}=8$