Question

If $m^{th}$ term of HP is n while $n^{th}$ term is m find its ${(m+n)}^{th}$ term

Answer 1

Given $T_m=nor\frac{1}{a+(m-1)d}=n$ where a is the first term and d is the common difference of the corresponding AP
So $a+(m-1)d=\frac{1}{n}$

and $a+(n-1)d=\frac{1}{m}$

$\Rightarrow (m-n)d=\frac{m-n}{mn}$

$d=\frac{1}{mn}$

So $a=\frac{1}{n}-\frac{(m-1)}{mn}=\frac{1}{mn}$

Hence $T_{(m+n)}=\frac{1}{a+(m+n-1)d}=\frac{mn}{1+m+n-1}$
= $\frac{mn}{m+n}$