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In Geometric progression the sum of first of three terms is 28 and the product is 512 find those numbers

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    a+ar+ar^2=28
    a^3 * r^3=512

    one the second equation, take the cube root of each side.
    ar=cubrt(512)=8

    a+8+ar^2=28
    a(1+r^2)=20

    but a= 8/r
    8(1+r^2)=20r
    8r^2-20 r+8=0

    r=(20)+-sqrt(400-256) /16

    r=20+- sqrt(144) /16 = (20+-12)/16=2 or 1/2

    then a= 4 or 16

    check a=4, r=2

    4 + 8 + 16=28
    a^3 r^3=64*8=512 checks, a=4, r=2

    check a=8, r=1/2
    8 +4 + 2 does not check.

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