Question

If the pth term of an A.P. is q and the qth term is p, then find its rth term.

Answer 1

Given that, $t_p=q$

$\therefore a+(p-1)d=q.....................\left(i\right)$

also given that, $t_q=p$

$\therefore a+(q-1)d=p.....................\left(ii\right)$

Subtracting $\left(ii\right)$ from $\left(i\right)$, we get,

$d(p-q)=q-p$

$Hence,d=-1$

$\therefore a-p+1=q$

$\Rightarrow a=p+q-1$

Now, we know that $t_r=a+(r-1)d$

$\therefore t_r=p+q-1+(r-1)(-1)$

$\Rightarrow t_r=p+q-1-r+1$

$\therefore t_r=p+q-r$

Hence, the rth term $\left(t_r\right)=p+q-r$