Question

Let $t_r$ be the r${}^{th}$ term of an A.P, for $r=1,2,3,......$ for some positive integer $m$, $n$, $t_m=\frac{1}{n}$ & $t_n=\frac{1}{m}$ then $t_{mn}$ equals

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

Answer 1

The correct option is A $1$

Let $a$ be the first term and $d$ is a common difference of given A.P

nth term of an A.P is given by, $a_n$ $=$ $a+(n-1)d$

given, $t_m$ $=$ $a+(m-1)d$ $=$ $\frac{1}{n}$

$\Rightarrow$ $a-d$ $=$ $\frac{1}{n}-md$ $\to$ (1)

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

$\Rightarrow$ $a-d$ $=$ $\frac{1}{m}-nd$ $\to$ (2)

solving (1) and (2),

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

so $t_{mn}$ $=$ $a+(mn-1)d$

$=a-d+mnd$

$=\frac{1}{m}$ $-nd$ $+1$

$=$ $\frac{1}{m}$ - $\frac{1}{m}$ $+1$

$=$ $1$, [Using above relations]