Question

If $m^{th}$ term of an A.P. is $n$ and $n$th term is $m$, then the $r^{th}$ term is

• A. $m+n-r$
• B. $m+n+r$
• C. $m-n+r$
• D. $r-(m+n)$

Answer 1

The correct option is A $m+n-r$

The $r^{th}$ term of an A.P with the first term as ${}^{\prime }{a}^{\prime }$ and common difference ${}^{\prime }{d}^{\prime }$, is

$a_r=a+(r-1)d$.

Hence

$a+(m-1)d=n$...........$\left(i\right)$

$a+(n-1)d=m$ ..........$\left(ii\right)$

Subtracting $\left(ii\right)$ from $\left(i\right)$ gives us

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

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

$d=-1$

Hence

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

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

$a=n+m-1$.

Hence

$a_r=a+(r-1)d$

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

$=n+m-1-r+1$

$=n+m-r$

Hence

$a_r=n+m-r$.