Question

If the $m^{th}$ term of a H.P. be equal to $n$, and the $n^{th}$ term be equal to $m$, prove that the $(m+n{)}^{th}$ term is equal to $\frac{mn}{m+n}$.

Answer 1

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

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

Solving the above, we get $a=\frac{1}{mn}$ and $d=\frac{1}{mn}$

$\therefore (m+n{)}^{\text{th}}$ term of the HP is $\frac{1}{a+(m+n-1)d}=\frac{1}{\frac{1}{mn}+\frac{m+n-1}{mn}}=\frac{mn}{m+n}$