In an A.P, if the p^th term is 1/q and q^th term is 1/p, prove that the sum of first pq terms is 1/2 (pq + 1), where p ≠ q.

Solution:

It is known that general term of an A.P is

a_n = a + (n - 1) d

p^thterm = a_n = a + (n - 1) d = 1/q ....(1)

q^thterm = a_n = a + (n - 1) d = 1/p ....(2)

Subtracting (2) from (1) , we obtain

⇒ ( p - 1) d - (q - 1) d = 1/q - 1/p

⇒ ( p -1- q +1) d = (p - q)/pq

⇒ ( p - q) d = (p - q)/pq

⇒ d = 1/pq

Putting the value of d in (1), we obtain

⇒ a + ( p - 1) 1/pq = 1/q

⇒ a = 1/q - 1/q + 1/pq = 1/pq

Therefore,

S_pq = pq/2 [2a + (pq - 1) d]

S_pq = pq/2 [2/pq + (pq - 1) 1/pq]

= pq/2 [2/pq + 1 - 1/pq]

= pq/2 [1/pq + 1]

= 1/2 + pq/2

= 1/2 (pq + 1)

Thus, the sum of the first pq terms of the A.P is 1/2 (pq + 1)

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NCERT Solutions Class 11 Maths Chapter 9 Exercise 9.2 Question 5

In an A.P, if the p^th term is 1/q and q^th term is 1/p, prove that the sum of first pq terms is 1/2 (pq + 1), where p ≠ q

Summary:

Given the above conditions, we have proven that the sum pq terms in the series are 1/2 (pq + 1)