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Deepak Bora
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If the pth term of an AP is q and its qth term is p then show that is (p+q)th term is zero.

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Question of RS Aggarwal book based on Arithmetic Progression For ICSE & CBSE Board Students.
Here the two nth term of AP is given you have to show that its both nth term Addition is Zero.
Question Number 26 Of Exercise 11A of RS Aggarwal Solution.

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  1. Let us consider that the first term of the AP is a and the common ratio is d.

    Given,

    P-th term = Q

    => a + (P – 1)d = Q …..(i)

    and

    Q-th term = P

    => a + (Q – 1)d = P …..(ii)

    We have

    a + (P – 1)d = Q …..(i)

    a + (Q – 1)d = P …..(ii)

    On subtraction,
    we get

    (P – 1 – Q + 1)d = Q – P

    => (P – Q)d = -(P – Q)

    => d = -1 [eliminating (P – Q)]

    So,
    common ratio (d) = -1

    Putting d = -1 in (i),
    we get

    a + (P – 1)(-1) = Q

    => a = P + Q – 1

    So,
    first term = P + Q – 1

    Therefore, the (P + Q)-th term is

    = a + (P + Q – 1)d

    = P + Q – 1 + (P + Q – 1)(-1)

    = P + Q – 1 – P – Q + 1

    = 0 [Proved]

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