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CHRIST MATH

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Divide 20 into 4 parts which are in AP such that ratio between the product of the 1st part and the 4th part to the 2nd and 3rd part is 2:3 find the AP

  • CHRIST MATH - ,

    What does AP mean and what does this have to do with Christ?

  • CHRIST MATH - ,

    I will assume that AP stands for arithmetic progression

    so you are given:
    a + (a+d) + (a+2d) + (a+3d) = 20
    4a + 6d = 20
    2a + 3d = 10

    product of 1st and 4th = a(a+3d)
    product of 2nd to 3rd = (a+d)(a+2d)

    then:
    a(a+3d)/( (a+d)(a+2d) ) = 2/3
    (a^2 + 3ad)/(a^2 + 3ad + 2d^2) = 2/3
    3a^2 + 9ad = 2a^2 + 6ad + 4d^2 = 0
    a^2 + 3ad - 4d^2 = 0
    (a-d)(a+4d) = 0

    a = d or a = -4d

    case1: a = d
    in 2a+3d=10
    5d = 10,
    d = 2, so a=d = 2
    the terms are 2, 4, 6 , 8

    case2: a = 4d
    in 2a + 3d = 10
    -4d + 3d = 10
    -d = 10
    d = -10, then a = 40
    the terms are 40, 30, 20, 10

  • typo - CHRIST MATH - ,

    I just noticed a typo in my second case
    It should say:

    case2: a = -4d
    in 2a + 3d = 10
    -8d + 3d = 10
    -2d = 10
    d = -5, then a = 20
    the terms are 20 , 15, 10 , 5

    but these do not add up to 20 , even though the ratio of product property works

    so the only solution are the terms

    2, 4, 6, 8

    notice they add up to 20
    and (2x8)/(4x6) = 16/24 = 2/3

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