Question

Let $a_1,a_2,a_3,.............$ be in harmonic progression with $a_1=5and{a}_{20}=25.$ The least positive integer n for which $a_n<0$ is

• A. 22
• B. 23
• C. 24
• D. 25

Answer 1

The correct option is D

25

If $a_1,a_2,a_3,......$ are in harmonic progression, then $\frac{1}{a_1},\frac{1}{a_2},\frac{1}{a_3}.....$ are in AP.
First term of AP, $\frac{1}{a_1}=\frac{1}{5}$
20th term of AP, $\frac{1}{a_{20}}=\frac{1}{25}$
$\Rightarrow \frac{1}{5}+19d=\frac{1}{25}$
$\Rightarrow d=\frac{-4}{19\times 25}$
We have to find the least positive integer n for which $a_n<0$
$\Rightarrow \frac{1}{5}+(n-1)d<0$
$\Rightarrow \frac{1}{5}+(n-1)\left(\frac{-4}{19\times 25}\right)<0$
$\Rightarrow (n-1)>\frac{95}{4}$
$\Rightarrow n>24.75$