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Question

The sum of p terms of an A.P. is q, and the sum of q terms is p; find the sum of p+q terms.

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Solution

Let the first term of the AP be a and the common difference be d
Sum of p terms =q

q=pa+p(p1)2d ........... Since, sum of n terms =na+n(n1)2d

2qp=2a+(p1)d ...... (i)

Similarly, sum of q terms =p=qa+q(q1)2d

2pq=2a+(q1)d ....... (ii)

Subtract (i)(ii)

2(q2p2pq)=(pq)d

2((qp)(q+p)pq)=(pq)d

d=2p+qpq ...... (iii)

Sum of first p+q terms = (p+q)a+(p+q)(p+q1)2d

= (p+q)2(2a+(p1)d+qd)

Using (i) and (iii),

=(p+q)2(2qp+q(2p+qpq))

=(p+q)×(qpp+qp)

=(p+q)×(1)

=(p+q)

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