Question

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?

Answer 1

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

Answer 2

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

Answer 3

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