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 which $a_n<0$

• A. 22
• B. 23
• C. 24
• D. 25

Answer 1

Answer: (D)

$a_1,a_2,a_3,$ are in $H.P.$
$\Rightarrow \frac{1}{a_1},\frac{1}{a_2},\frac{1}{a_3},\dots$ are in A.P.
$\Rightarrow \frac{1}{a_n}=\frac{1}{a_1}+(n-1)d<0,$
where $\frac{\frac{1}{25}-\frac{5}{25}}{19}=d=\left(\frac{-4}{9\times 25}\right)$
$\Rightarrow \frac{1}{5}+(n-1)\left(\frac{-4}{19\times 25}\right)<0$
$\frac{4(n-1)}{19\times 5}>1$
$n-1>\frac{19\times 5}{4}$
$n>\frac{19\times 5}{4}+1\Rightarrow n\ge 25.$