Question

If the $7^{\text{th}}$ and ${12}^{\text{th}}$ terms of a H.P. are $\frac{1}{10}$ and $\frac{1}{25}$ respectively, then the ${20}^{\text{th}}$ term of H.P. is

• A. $\frac{1}{37}$
• B. $\frac{1}{42}$
• C. $\frac{1}{45}$
• D. $\frac{1}{49}$

Answer 1

The correct option is D $\frac{1}{49}$

We know that, if $a_1,a_2,a_3,...$ are in H.P., then $\frac{1}{a_1},\frac{1}{a_2},\frac{1}{a_3},...$ are in A.P.
Let $a$ be the first term and $d$ be the common difference of corresponding A.P., then
$T_7=a+6d=10...\left(1\right)$
$T_{12}=a+11d=25...\left(2\right)$
From $\left(1\right)$ and $\left(2\right),$ we have
$d=3,a=-8$
$\therefore T_{20}=a+19d$
$=-8+19\times 3=49$

Hence, ${20}^{\text{th}}$ term of H.P. $=\frac{1}{49}$