Question

if Pth term of A.P be 1/9 and 9th term be 1/p . prove that sum of 1st pq term must be 1/2 ​

Answer 1

Given pth term = 1/q

That is ap = a + (p - 1)d = 1/q

aq + (pq - q)d = 1  --- (1)

Similarly, we get ap + (pq - p)d = 1  --- (2)

From (1) and (2), we get

aq + (pq - q)d = ap + (pq - p)d  

aq - ap = d[pq - p - pq + q]

a(q - p) = d(q - p)

Therefore, a = d

Equation (1) becomes,

dq + pqd - dq = 1  

d = 1/pq

Hence a = 1/pq

Consider, Spq = (pq/2)[2a + (pq - 1)d]

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

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

                   = (1/2)[pq + 1]