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Q.
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-
Sequences and Series
Solution:
$m^{\text {th }}$ 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}{m n} $
$\Rightarrow d=\frac{1}{m n}$
$a=\frac{1}{n}-\frac{m-1}{m n}=\frac{1}{m n}$
$\therefore r^{\text {th }}$ term of A.P. is
$T_r=\frac{1}{m n}+\frac{r-1}{m n}=\frac{r}{m n}$
$\therefore $ rth term of H.P. $=\frac{m n}{r}$