#### Question

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

#### Solution

For an A.P.

`t_m = n`

`=> a + (m - 1)d = n` .....(i)

And `t_n = m`

=> a + (n - 1)d = m .....(ii)

Substracting (i) from (ii) we get

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

=> d(n - m) = m - n

=> -d(n - m) = m - n

=> d = -1

Substituting d = -1 in equation (i) we get

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

=> a - m + 1 = n

=> a = m + n -1

Now `t_r = a + (r - 1)d`

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

= m + n - 1 - r + 1

= m + n - r

Is there an error in this question or solution?

Solution In an A.P, If Mth Term is N and Nth Term is M, Show that Its Rth Term is (M + N – R) Concept: Arithmetic Progression - Finding Their General Term.