Question

Five numbers are in H.P. The middle term is 1 and the ratio of the second and the fourth terms is 2:1. Then the sumof the first three terms is

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

Answer 1

Answer: (A)

$\frac{11}{2}$

Let $(a-2d),(a-d),a,(a+d),(a+2d)$; be five terms in A.P.

So, our five terms in H.P. will be

$\frac{1}{a-2d};\frac{1}{a-d};\frac{1}{a};\frac{1}{a+d};\frac{1}{a+2d}$

Since, middle term is 1,

$\therefore \frac{1}{a}=1$ Thus, $a=1$.

According to problem:

Ratio of second and fourth terms is 2:1.

Thus, $\frac{\frac{1}{a-d}}{\frac{1}{1+d}}=\frac{2}{1}$

So,

$\frac{a-d}{a+d}=\frac{1}{\frac{2}{1}}=\frac{1}{2}$

$\Rightarrow \frac{1-d}{1+d}=\frac{1}{2}$

$\therefore d=\frac{1}{3}$

$\therefore$ First term $=\frac{1}{1-\frac{2}{3}}=3$

Second term $=\frac{1}{1-\frac{1}{3}}=\frac{3}{2}$

Third term $=1$

$\therefore Sum=3+\frac{3}{2}+1=\frac{11}{2}$