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 $T_{mn}$ =

Answer 1

We will proceed the same way we did for other question. The only difference is that we have variables which are also the index (m and n). We will calculate the common difference and then the first term, if necessary.

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

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

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

$=\frac{m-n}{mn}$

$m\ne n$

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

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

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

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

$[a=\frac{1}{n}-(m-1)d,from(1\left)\right]$

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

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

=1