Question

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

Answer 1

For an A.P.

t_m = n

`=>` a + (m – 1)d = n  ...(i)

And t_n = m

`=>` a + (n – 1)d = m  ...(ii)

Subtracting (i) from (ii), we get

(n – 1)d – (m – 1)d = m – n

`=>` nd – d – md + d = m – n

`=>` d(n – m) = m – n

`=>` – d(m – n) = 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