If the sum of the first p terms of an AP is the same as the sum of its first q terms (where pq) then show that the sum of its first (p+q) terms is zero.


Answer:


Step by Step Explanation:
  1. We know that the sum of first n terms of an AP is given by Sn=n2(2a+(n1)d), where a is the first term and n is the number of terms in the AP.
  2. We are given that Sp=Sqp2(2a+(p1)d)=q2(2a+(q1)d)p(2a+(p1)d)=q(2a+(q1)d)2ap+(p1)dp=2aq+(q1)dq2ap2aq=(q1)dq(p1)dp2a(pq)=q2ddqp2d+dp2a(pq)=q2dp2d+dpdq2a(pq)=d(p2q2)+d(pq)2a(pq)=d(pq)(p+q)+d(pq)2a(pq)=(pq)[d(p+q)+d]2a=d(p+q)+d2a=(1pq)d(i)
  3. Now, the sum of first (p+q) terms of the given AP is Sp+q=p+q2(2a+(p+q1)d)=p+q2((1pq)d+(p+q1)d) [Using(i)] =p+q2(dpdqd+pd+qdd)=p+q2(0)=0
  4. Hence, the sum of (p+q) terms is 0 .

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