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Algebra/Number Theory

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In a sequence of four positive numbers, the first three are in geometric progression and the last three are in arithmetic progression. The first number is 12 and the last number is 452. The sum of the two middle numbers can be written as ab where a and b are coprime positive integers. Find a+b.

  • Algebra/Number Theory -

    The numbers are
    12, 12+d, 12+2d, 452
    12, 12+d, (12+d)r, (12+d)r^2=452

    (12+d)r^2 = 452
    12+2d = (12+d)r

    so,

    r = (12+2d)/(12+d)

    (12+d)((12+2d)/(12+d))^2 = 452
    (12+2d)^2 = 452(12+d)
    d = 112.71
    r = 1.09377

    The numbers are

    12, 124.71, 237.42, 452

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