Question

If $m^{th}$ term of H.P. is $n$ and $n^{th}$ term is $m,$ find its ${(m+n)}^{th}$ term.

Answer 1

Given $T_m=n$ or $\frac{1}{a+(m-1)d}=n;$ where $a$ is the first term and $d$ is the common difference of the corresponding A.P.
so $a+(m-1)d=\frac{1}{n}$
and $a+(n-1)d=\frac{1}{m}$
$\Rightarrow$ $(m-n)d=\frac{m-n}{mn}$
or $d=\frac{1}{mn}$
so $a=\frac{1}{n}-\frac{(m-1)}{mn}=\frac{1}{mn}$
Hence $T_{(m+n)}=\frac{1}{a+(m+n-1)d}=\frac{mn}{1+m+n-1}=\frac{mn}{m+n}$