Sequences And Series Question 15
Question 15 - 31 January - Shift 1
Let $a_1, a_2, \ldots \ldots, a_n be$ in A.P. If $a_5=2 a_7$ and $a _{11}=18$, then
$12(\frac{1}{\sqrt{a _{10}}+\sqrt{a _{11}}}+\frac{1}{\sqrt{a _{11}}+\sqrt{a _{12}}}+\ldots . \frac{1}{\sqrt{a _{17}}+\sqrt{a _{18}}})$ is equal to _________
Show Answer
Answer: 8
Solution:
Formula: General Term of an AP
$2 a_7=a_5$ (given)
$2(a_1+6 d)=a_1+4 d$
$a_1+8 d=0$
$a_1+10 d=18$
By (1) and (2) we get $a_1=-72, d=9$
$a _{18}=a_1+17 d=-72+153=81$
$a _{10}=a_1+9 d=9$
$12(\frac{\sqrt{a _{11}}-\sqrt{a _{10}}}{d}+\frac{\sqrt{a _{12}}-\sqrt{a _{11}}}{d}+\ldots . . \frac{\sqrt{a _{18}}-\sqrt{a _{17}}}{d})$
$12(\frac{\sqrt{a _{18}}-\sqrt{a _{10}}}{d})=\frac{12(9-3)}{9}=\frac{12 \times 6}{6}=8$
