In an A.P., if mth term is n and nth term is m, show that its rth term is ( m+n-r)

AP

we have

a_{m} = a + (m - 1)d = n…(1)

a_{n} = a + (n - 1)d = m…(2)

from (1) and (2) we have

(m -n)d =n - m

d = - 1. . . . . . ( 3)

and a = n + m - 1

Now for rth term we have

a_{r} = a + (r - 1)d

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

= n + m - r

Hence rth term will be n + m - r