In an A.P. if mth term is n and the nth term is m, where m ≠ n, find…

The correct answer is Option (4) → $n+m-p$

Let the A.P. have first term a and common difference d.

Given:

$T_m = a + (m-1)d = n \quad (1)$

$T_n = a + (n-1)d = m \quad (2)$

Subtract (1) from (2):

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

Substitute $d=-1$ into (1):

$a + (m-1)(-1) = n \Rightarrow a - m + 1 = n \Rightarrow a = n + m – 1$

Now, the pth term:

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

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

$=n+m−p$