Question

Let $a_{1}, a_{2}, a_{3}, \ldots . .$ 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 $H.P.$
$\Rightarrow \frac{1}{a_{1}}, \frac{1}{a_{2}}, \frac{1}{a_{3}}, \ldots$ are in A.P.
$\Rightarrow \frac{1}{a_{n}}=\frac{1}{a_{1}}+(n-1) d<0,$
where $\frac{\frac{1}{25}-\frac{5}{25}}{19}=d=\left(\frac{-4}{9 \times 25}\right)$
$\Rightarrow \frac{1}{5}+(n-1)\left(\frac{-4}{19 \times 25}\right)<0$
$\frac{4(n-1)}{19 \times 5}>1$
$n-1>\frac{19 \times 5}{4}$
$n>\frac{19 \times 5}{4}+1 \Rightarrow n \geq 25 .$