Question
In an A.P. if $m^{th}$ term is n and the $n^{th}$ term is m, where m $\ne$ n, find the $p^{th}$ term.

Answer

We have $a_m=a+(m-1) d=n$,.....(i) and $a_n=a+(n-1) d=m$.....(ii)
Solving (i) and (ii), we obtain
$(m-n) d=n-m$, or $d=-1$,  .....(iii)
and $\mathrm{a}=\mathrm{n}+\mathrm{m}-1$.....(iv)
Thus, $a_p=a+(p-1) d$ $=n+m-1+(p-1)(-1)=n+m-p$
Thus, the $\mathrm{p}^{\text {th }}$ term is $\mathrm{n}+\mathrm{m}-\mathrm{p}$.

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