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.

• A. $m+n+p$
• B. $m-n+p$
• C. $m+n-p$
• D. $m-n-p$

Answer 1

Answer: (C)

$m+n-p$

We have the $n^{th}$ term of an AP,

$T_n=an=a+(n-1)d$

Where, $a$->first term and $d$-> common difference

For $m^{th}$ term, $am=a+(m-1)d=n$ ................(1)

$an=a+(n-1)d=m$ ..............(2)

on subtracting (1) and (2),

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

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

$(m-n)d=n-m$

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

Substitute in equation (1),

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

$a-m+1=n$

$a=n+m-1$

Term $p$ is,

$ap=a+(p-1)d$

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

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

$=n+m-p$ Option C.