Question

Let $a_1,a_2,a_3,\dots$ be in harmonic progression with $a_1=5$ and $a_{20}=25$. The least positive integer $n$ for whlch $a_n<0$ ls

• A. 22
• B. 23
• C. 24
• D. 25

Answer 1

Answer: (D)

25

$a_1,a_2,a_3......$ are in H.P.

$\Rightarrow \frac{1}{a_1},\frac{1}{a_2},\frac{1}{a_3}.....$ are in A.P.

$\Rightarrow \frac{1}{a_n}=\frac{1}{a_1}+(n-1)d$

$\Rightarrow \frac{1}{a_{20}}=\frac{1}{a_1}+19d$

$\Rightarrow d=\frac{\frac{1}{25}-\frac{5}{25}}{19}$

$=\left(\frac{-4}{9\times 25}\right)$

Given that $a_n<0$

$\Rightarrow \frac{1}{a_n}<0$

$\Rightarrow \frac{1}{a_1}+(n-1)\left(\frac{-4}{9\times 25}\right)<0$

$\Rightarrow \frac{2(n-1)}{19\times 5}>1$ $[\because \frac{1}{a_n}=5]$

$\Rightarrow (n-1)>\frac{19\times 5}{4}$

$\Rightarrow n>\frac{19\times 5}{4}+1$

$\Rightarrow n⩾24.75\approx 25$

$\Rightarrow n⩾25$