Question

The first two term of an H.P, are $\frac{2}{5}$ and $\frac{12}{23}$. The value of the largest term of the H.P. is

• A. $\frac{72}{73}$
• B. $6$
• C. $\frac{1}{6}$
• D. None of these

Answer 1

The correct option is B $6$

Let a, b be the first two terms of an AP.

Then first two terms of HP is

$\frac{1}{9}=2/5$ $\Rightarrow a=\frac{5}{2}$

and $\frac{1}{b}=\frac{12}{23}$ $\Rightarrow b=\frac{23}{12}$

$\therefore$ common difference, $d=b-a$

$=\frac{23}{12}-\frac{5}{2}$

$=\frac{23-30}{12}$

$=\frac{-7}{12}$

Let $n^{th}$ term be the smallest positive term of the AP

$\therefore a_n\ge 0$

$\Rightarrow a+(n-1)d\ge 0$

$\Rightarrow \frac{5}{2}+(n-1)\left(\frac{-7}{12}\right)\ge 0$

$\Rightarrow 30-7(n-1)\ge 0$

$\Rightarrow 30-7n+7\ge 0$

$\Rightarrow -7n+37\ge 0$

$\Rightarrow -7n\ge -37$

$\Rightarrow 7n\le 37$

$\Rightarrow n\le \frac{37}{7}$

$\Rightarrow n\le 5.3$ (approx.)

$\therefore n=5$ since n is a natural number.

$\therefore a_n=a_5$

$=a+(5-1)d$

$=\frac{5}{2}+4\cdot \left(\frac{-7}{12}\right)=\frac{5}{2}-\frac{7}{3}=\frac{15-14}{6}=\frac{1}{6}$

$\therefore$ Value of largest term of HP$=\frac{1}{a_5}$

$=\frac{1}{1/6}$

$=6$.