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

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We have $a_{m}=a+(m-1) d=n,$        ......$(1)$

and $\quad a_{n}=a+(n-1) d=m$          .........$(2)$

Solving $(1)$ and $(2),$ we get

$(m-n) d=n-m,$ or $d=-1,$          ...........$(3)$

and $\quad a=n+m-1$             ...........$(4)$

Therefore $\quad a_{p}=a+(p-1) d$

$=n+m-1+(p-1)(-1)=n+m-p$

Hence, the $p^{\text {th }}$ term is $n+m-p$

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