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11 January, 00:19

5. In an A. P, the sum of three consecutive terms is 24. When 1 is subtracted to the first term, 2 to the

second, then the terms form a G. P. Find the terms of the A. P.

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  1. 11 January, 03:58
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    Step-by-step explanation:

    Let the three numbers be a-d, a, a+d being a series in AP.

    The sum of the three numbers is 24 = a-d + a + a+d = 3a

    Thus a = 8.

    Now a-d-1, a-2, a+d form a GP. It means, the common ratios should be equal or the middle term should be the Geometric Mean of the first and third terms, or

    (a-d-1) (a+d) = (a-2) ^2 or

    (8-d-1) (8+d) = (8-2) ^2, or

    (7-d) (8+d) = 36, or

    56 - 8d + 7d - d^2 = 36, or

    56 - 8d + 7d - d^2 - 36 = 0, or

    d^2 + d - 20 = 0

    (d+5) (d-4) = 0

    Thus d = - 5 or 4

    So the AP is 13,8,3 or 4,8,12

    Check: 13,8,3 in AP becomes (13-1), (8-2), 3 = 12, 6, 3 which is a GP with r = (1/2)

    4,8,12 in AP becomes (4-1), (8-2), 12 = 3, 6, 12 which is a GP with r = 2
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