Question

If the $m^{\text{th }}$ term of a H.P. be $n$ and $n^{\text{th }}$ term be $m$, then the $r^{\text{th }}$ term will be-

• A. $\frac{r}{mn}$
• B. $\frac{mn}{r+1}$
• C. $\frac{mn}{r}$
• D. $\frac{mn}{r-1}$

Answer 1

Answer: (C)

$m^{\text{th }}$ term of H.P. is $n$
so $m^{\text{nh }}$ term of A.P. is $\frac{1}{n}$
$n^{\text{th }}$ term of H.P. is $m$
$n^{\text{th }}$ term of A.P. is $\frac{1}{m}$
Let first term of A.P. is a and common difference is $d$.
$\therefore t_m=a+(m-1)d=\frac{1}{n}$
$t_n=a+(n-1)d=\frac{1}{m}$
$\therefore (m-n)d=\frac{m-n}{mn}$
$\Rightarrow d=\frac{1}{mn}$
$a=\frac{1}{n}-\frac{m-1}{mn}=\frac{1}{mn}$
$\therefore r^{\text{th }}$ term of A.P. is
$T_r=\frac{1}{mn}+\frac{r-1}{mn}=\frac{r}{mn}$
$\therefore$ rth term of H.P. $=\frac{mn}{r}$