Question

Let $T_r$ be the $r^{th}$ term of an A.P for $r=1,2,3,...$ if for some positive integers $m,n$, we have $T_m=\frac{1}{n}$ and $T_n=\frac{1}{m}$, then $T_{mn}$ equals

• A. $\frac{1}{mn}$
• B. $\frac{1}{m}+\frac{1}{n}$
• C. $1$
• D. $0$

Answer 1

Answer: (C)

$1$

Given $T_m=\frac{1}{n}$

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

Also, $T_n=\frac{1}{m}$

$\Rightarrow a+(n-1)d=\frac{1}{m}$ ....(2)

Subtracting equations (1) and (2), we get

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

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

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

Now, $a=\frac{1}{n}-\frac{m-1}{mn}$

$\Rightarrow a=\frac{1}{mn}$

Consider, $T_m=a+(mn-1)d$

$\Rightarrow T_{mn}=\frac{1}{mn}+\frac{(mn-1)}{mn}$

$\Rightarrow T_{mn}=\frac{mn}{mn}$

$\Rightarrow T_{mn}=1$