Question

if the pth term of an A.P. is 1/q and the qth term is is 1/p , prove that the sum of the first pq terms is 1/2[pq+1]​

Answer 1

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Given pth term = 1/q

a + (p - 1)d = 1/q

aq + (pq - q)d = 1  --- (1)

Similarly, 

 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]

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Answer 2

Given pth term = 1/q

a + (p - 1)d = 1/q

aq + (pq - q)d = 1  --- (1)

Similarly, 

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]

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