Question

If mth term of an A.P. is n and nth term is m, then write its pth term.

Answer 1

We have,

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

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

and, $a_n=a+(n-1)d=n-m$

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

Subtracting equation (ii) from equation (i), we get

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

$\Rightarrow (m-1)d-(n-1)d=n-m$

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

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

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

$\Rightarrow d=-1$

Putting d = -1 in equation (i), we get

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

$\Rightarrow a-m+1=n\Rightarrow a=n+m-1$

Now,

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

$=n+m-1+(p-1)(-1)$ $[\because a=n+m-1]\text{and}d=-1$

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

$a_p=m+n-p$

Hence pth term is m + n - p