Question

If the pth term of an A.P. is x and qth term is y, show that the sum of (p + q) terms is (p + q)/2[x + y + (x - y/p - q)].

Answer 1

Given: a_p = x and a_q = y

We know that,

a_n = a + (n – 1)d

a_p = a + (p – 1)d

⇒ x = a + (p – 1)d …(i)

Now,

a_q = a + (q – 1)d

⇒ y = a + (q – 1)d …(ii)

From eq. (i) and (ii), we get

x – (p – 1)d = y – (q – 1)d

⇒ x – y = (p – 1)d – (q – 1)d

⇒ x – y = d [p – 1 – q + 1]

⇒ x – y = d[ p – q]

⇒ d = (x - y)/(p - q) …(iii)

Adding, Eq (i) and (ii), we get

x + y = 2a + (p – 1) + (q – 1)d

⇒ x + y = 2a + d[p + q – 1 – 1]

⇒ x + y = 2a + d (p + q – 1) –d

⇒ x + y + d = 2a + (p + q – 1)d …(iv)

We know that,

Hence Proved