Question

Five numbers are in HP. The middle term is $1$ and the ratio of the second and the fourth terms is $2:1$. Then, the sum of the first three terms is

• A. $\frac{11}{2}$
• B. $5$
• C. $2$
• D. $\frac{14}{3}$

Answer 1

The correct option is A $\frac{11}{2}$

Let $a-2d,a-d,a+d,a+2d$ are in AP
then $\frac{1}{a-2d},\frac{1}{a-d},\frac{1}{a+d},\frac{1}{a+2d}$ are in HP
Given, middle term $=1$
$\Rightarrow$ $\frac{1}{a}=1$
and $\frac{\text{Second term}}{\text{First term}}=\frac{2}{1}$
$\Rightarrow$ $\frac{1}{a-d}\times (a+d)=\frac{2}{1}$
$\Rightarrow$ $a=3d$ $\Rightarrow$ $d=\frac{1}{3}$ ($\because a=1$)
$\therefore$ Sum of first three terms
$=\frac{1}{1-\frac{2}{3}}+\frac{1}{1-\frac{1}{3}}+1=3+\frac{3}{2}+1=\frac{11}{2}$