Geometric Progression.

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The fifth of an exponential sequence (GP) is greater than the fourth term by 13.5 and fourth term is greater than the thrid by 9.Find:common ratio and the first term.

  • Geometric Progression. -

    (x + 9) / x = (x + 22.5) / (x + 9)

    x^2 + 18 x + 81 = x^2 + 22.5 x

    81 = 4.5 x

    a1, a2, 18, 27, 40.5

  • Geometric Progression. -

    ar^4 - ar^3 = 13.5
    ar^3(r - 1) = 13.5 **
    ar^3 - ar^2 = 9
    ar^2(r-1) = 9 ***

    divide ** by ***
    r = 13.5/9 = 1.5
    sub into ***
    a(1.5)^2 (1.5-1) = 9
    a(1.125) = 9
    a = 9/1.125 = 8

    first term is 8, common ratio = 1.5 or 3/2

    check:
    the terms are:
    8, 12, 18, 27, 40.5, ...
    term5 - term4 = 40.5-27 = 13.5 , good!
    term4 - term3 = 27-18 = 9 , as it should be!

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