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Question

If the sum of first p terms of an A.P.is equal to the sum of the first q terms, then find the sum of the first (p+q) terms.

Solution
Verified by Toppr

Let a and d be the first term and the common difference of the A.P. respectively.
Thus, Sp=p2[2a+(p1)d]
and Sq=q2[2a+(q1)d]
Now according to the given condition,
p2[2a+(p1)d]=q2[2a+(q1)d]p[2a+(p1)d]=q[2a+(q1)d]2ap+pd(p1)=2aq+qd(q1)2a(pq)+d[p(p1)q(q1)]=02a(pq)+d[p2pq2+q]=02a(pq)+d[(pq)(p+q)(pq)]=02a(pq)+d[(pq)(p+q1)]=02a+d(p+q1)=0d=2ap+q1Sp+q=p+q2[2a+(p+q1).d]=p+q2[2a+(p+q1)(2ap+q1)]=0

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