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March 25, 2017

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An operation produces (A-1)/A from a fraction A=m/n, where m is not equal with n and m is not zero. If the initial value of A is 22/47 and the operation is repeated 2012 times, the final output is a/b. What is a and b?

  • Maths - ,

    A1 = 22/47
    A2 = (22/47 - 1)/(22/47
    = -25/22
    A3 = (-25/22 - 1)/(-25/22)
    = 47/25
    A4 = (47/25 - 1)/(47/25)
    = 22/47
    A5 = -25/22
    A6 = 47/25
    A7 = 22/47
    ...

    looks like if the subscript is divisible by 4, the answer is 22/47
    since 2012 divides evenly by 4, we get

    22/47 as a/b, so a = 22, b = 47

  • Maths - ,

    f(A) = 1 - 1/A
    so,
    f(m/n) = (m/n - 1)/(m/n)
    = (m-n)/m
    f((m-n)/m) = (m-n-m)/(m-n) = -n/(m-n)
    f(-n/(m-n)) = (-n+n-m)/(-n) = m/n

    so, f(f(f(A))) = A
    every 3rd iteration, we are back to A.

    2012/3 = 670 with remainder 2.
    Thus, f2012(A) = f2(A)

    so, take a look:
    f(22/47) = 1 - 47/22 = -25/22
    f(-25/22) = 1 + 22/25 = 47/25

  • Steve is right - Maths - ,

    Steve is right, it is a cycle of 3 terms, not 4 terms like I hastily concluded.
    so -25/22

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