Question

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

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

Answer 1

The correct option is C

$\frac{mn}{r}$

Given $T_m$ = n, $T_n$ = m for H.P. therefore for the corresponding A.P. $m^{th}$ term = $\frac{1}{n}$, $n^{th}$ term = $\frac{1}{m}$

Let a and d be the first term and common difference of this A.P., then

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

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

Solving these, we get a = $\frac{1}{mn}$, d = $\frac{1}{mn}$

Now, $r^{th}$ term of corresponding A.P.

= $a+(r-1)d$ = $\frac{1}{mn}+(r-1)\frac{1}{mn}$ = $\frac{1+r-1}{mn}$ = $\frac{r}{mn}$

Therefore $r^{th}$ term of corresponding H.P. is $\frac{mn}{r}$

Note : Students should remember this question as a fact.