Question

If $m^{th}$ term of the $AP$ is $\frac{1}{n}$ and the $n^{th}$ term of the $AP$ is $\frac{1}{m}$ find the value of $m{n}^{th}$ term of the $AP$

Answer 1

Formula,

$t_n=a+(n-1)d$

given,

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

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

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

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

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

substitute value of d in eqn (1), we get,

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

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

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

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

$=1$.