Math.G.P

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The 1st term of a G.P is equal to x-2,the 3rd term is x+6 and the arithmetic mean of the 1st and 3rd terms stands in the ratio to the 2nd 5:3.find the value of x.

  • Math.G.P -

    arithmetic mean of 1st and 3d
    = (x-2 + x+6)/2
    = (2x + 4)/2
    = x+2

    let the 2nd be y
    then y/(x-2) = (x+6)/y
    y^2 = (x-2)(x-6)
    y = ± √((x-2)(x+6))

    mean:2nd = 5:3
    (x+2)/√((x-2)(x+6)) = 5/3
    square both sides and expand
    (x^2 + 4x + 4)/(x^2 + 4x - 12) = 25/9
    25x^2 + 100x - 300 = 9x^2 + 36x + 36
    16x^2 + 64x - 336 = 0
    x^2 + 4x - 21 = 0
    (x+7)(x-3) = 0
    x = -7 or x = 3

    check:
    if x = -7,
    the 3 terms are : -9, ±3, -1

    (only the sequence -9, -3, -1 works)

    if x = 3
    the 3 terms are :1, ±3, 9
    (only the sequence 1 , 3, 9 works)

    so x = -7 or x = 3

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