Question

If the $7^{th}$ term of HP is $\frac{1}{10}$ and the ${12}^{th}$ term is $\frac{1}{25}$, then the ${20}^{th}$ term is

• A. $\frac{1}{41}$
• B. $\frac{1}{45}$
• C. $\frac{1}{37}$
• D. $\frac{1}{49}$

Answer 1

The correct option is D

$\frac{1}{49}$

Explanation for Correct Option:

Step 1: Solving given value

$7^{th}$ term of HP = $\frac{1}{10}$

${12}^{th}$ term = $\frac{1}{25}$

$7^{th}$ term of AP = $10$

$\Rightarrow a+6d=10....\left(i\right)$

${12}^{th}$ term of AP = $25$

$\Rightarrow a+11d=25....\left(ii\right)$

Step 2: Solving equations (i) and (ii), we get

$5d=15$

$\begin{array}{rcl}\therefore d& =& \frac{15}{5}\\ & =& 3\end{array}$

Step 3: Substitute value of $d$ in equation (i)

$\Rightarrow a+6d=10$

$\Rightarrow a+18=10$

$\Rightarrow a=-8$

${20}^{th}$ term of AP = $a+19d$

$$\begin{gathered} =-8+19\times 3 \\ =-8+57 \end{gathered}$$

$=49$

Since the $n^{th}$ term of H.P =$\frac{1}{({\mathrm{n}}^{\mathrm{th}}\mathrm{term}\mathrm{of}\mathrm{the}\mathrm{corresponding}\mathrm{A}.\mathrm{P})}$

Therefore, ${20}^{th}$ term of HP = $\frac{1}{49}$.

Hence option (3) is the answer.