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The sum of second term and fourth term of G.P is 30 and difference between sixth term and second term is 90,how to find third term

  • math -

    first term --- a
    common ratio --- r

    The sum of second term and fourth term of G.P is 30 ---> ar + ar^3 = 30
    ar(1 + r^2) = 30 , #1
    difference between sixth term and second term is 90 --> ar^5 - ar = 90
    ar(r^4 - 1) = 90 , #2

    divide #2 by #1
    (r^4 - 1)/(1+r^2) = 3
    (r^2 + 1)(r^2 - 1)/(1+r^2) = 3
    r^2 - 1 = 3
    r = ±2

    in #1
    if r = 2
    2a(1+4) = 30

    if r=-2
    -2a(1+4) = 30
    a = -3

    third term = ar^2
    = 12 or -12

    for a=3, r=2
    sequence is 3 6 12 24 48 96
    sum of 2nd and fourth = 6+24 = 30
    diff between 6th and 2nd = 96 - 6 = 90

    for a=-3, r=-2
    sequence is -3 6 -12 24 -48 96
    same as above, the negative terms don't enter the picture.

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