MathematicsIf the sum of p terms of an AP is q and the sum of q terms is p, then the sum of p + q terms will be:

If the sum of p terms of an AP is q and the sum of q terms is p, then the sum of p + q terms will be:


  1. A
    -p+q
  2. B
    -p-q
  3. C
    p+q
  4. D
    p-q 

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    Solution:

    Given that sum of p terms of AP is q and sum of q terms is p.
    We have to find sum of p+q terms.
    The sum of  n terms of an A.P. with the first term a and the common difference d is,
    Sn=n2[2a+(n-1)d]   As per the given data, the sum of p is Sp = q
    ⇒p2[2a+(p−1) d] = q
    ⇒2ap + (p-1) pd = 2q……………. (1)
    Also, the sum of q is Sq = p
    ⇒q2[2a+(q-1)d]=p ⇒2aq + (q-1) qd =2p………………(2)
    Subtracting equation (2) from equation (1), we get,
    2ap+(p−1) pd−[2aq+(q−1)qd]=2q−2p
    ⇒2ap+(p−1) pd−2aq−(q−1) qd=2q−2p
    ⇒2a(p−q) +p2d−pd−q2d+qd=−2(p−q)
    ⇒2a(p−q) +p2d−q2d−(p−q) d=−2(p−q)
    ⇒2a(p−q) +(p2−q2) d−(p−q)d=−2(p−q)
    ⇒2a+(p+q) d−d=−2 [Cancelling (p−q) from both sides]
    ⇒2a+(p+q−1) d=−2……(3)
    Now, we have to find sum of (p+q) terms,
    Sp+q=(p+q)2[2a+(p+q-1)d] ⇒Sp+q=(p+q)2-2 [Using equation 3] ⇒Sp+q=-p-q Therefore, the sum of p+q terms is -p-q.
    Hence, option 2 is correct.
     

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