Q. 294.0( 6 Votes )

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Answer :

Given that ap = 1/q and aq = 1/p

We have to show that Spq = (pq + 1)/2.


We know that in an AP with first term a and common difference d, the nth term (or the general term) is given by an = a + (n – 1) d.


ap = a + ( p – 1) d and aq = a + (q – 1) d


1/q = a + (p – 1) d … (1)


1/p = a + (q – 1) d … (2)


Solving (1) and (2),


(a – a) + (pd – d – qd + d) = 1/q – 1/p


(p – q) d = 1/q – 1/p


(p – q) d = (p – q)/ pq


d = 1/pq


Substituting d value in (1),


a + (p – 1) (1/pq) = 1/q


a + 1/q – 1/pq = 1/q


a = 1/q – 1/q + 1/pq


a = 1/pq


We know that the sum of the first n terms of an AP is given by Sn = [n (2a + (n – 1) d)] / 2.


Now,


Spq = [(pq) (2(1/pq) + (pq – 1)) (1/pq)] / 2


Spq =


Spq = (pq/2) (1/pq + 1)


Spq = (pq + 1)/ 2


Ans. From the given conditions, we showed that Spq = (pq + 1)/2.


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