If the sum of the first p terms of an AP be q and the sum of its first q terms be p then show that the sum of its first (p+q) terms is (p+q).


Answer:


Step by Step Explanation:
  1. Let a be the first term and d be the common difference of the given AP. Then, Sp=qp2(2a+(p1)d)=q2ap+p(p1)d=2q(i) And, Sq=pq2(2a+(q1)d)=p2aq+q(q1)d=2p(ii)
  2. On subtracting (ii) from (i), we get [2ap+p(p1)d][2aq+q(q1)d]=2q2p2a(pq)+(p2pq2+q)d=2q2p2a(pq)+(p2q2)d(pq)d=2(pq)2a(pq)+(pq)(p+q)d(pq)d=2(pq)2a+(p+q)dd=22a+(p+q1)d=2(iii)
  3. Now, the sum of the first (p+q) terms of the AP is Sp+q=p+q2(2a+(p+q1)d)=p+q2(2)[Using (iii)]=(p+q)
  4. Thus, the sum of the first (p+q) terms is (p+q).

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