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?
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
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, it is a cycle of 3 terms, not 4 terms like I hastily concluded.
so -25/22
To solve this problem, let's first understand the given operation. The operation takes a fraction A=m/n and produces (A-1)/A.
Given that the initial value of A is 22/47 and the operation is repeated 2012 times, we need to find the final output fraction.
Let's start by applying the operation once to the initial value of A:
(22/47 - 1)/(22/47)
To simplify this expression, we need to find a common denominator. The common denominator in this case is 47:
[(22-47)/47]/(22/47)
Simplifying the numerator gives:
(-25/47)/(22/47)
To divide fractions, we can multiply the first fraction by the reciprocal of the second fraction:
(-25/47) * (47/22)
Simplifying further gives:
-25/22
So, after one operation, the value of A becomes -25/22.
Now, we need to repeat this operation 2012 times. Let's denote An as the value of A after n operations.
A1 = -25/22
A2 = ((A1-1)/A1) = ((-25/22)-1)/(-25/22)
Similarly, we can calculate An for 3, 4, ..., 2012. However, instead of calculating each value of An manually, let's try to find a pattern:
If we substitute An-1 into the expression for An, we get:
An = ((An-1-1)/An-1)
We can simplify this expression by multiplying both the numerator and denominator by An-1, which gives:
An = (An-1^2 - An-1)/An-1
Simplifying further, we get:
An = An-1 - 1
This equation allows us to easily calculate each value of An based on the previous value:
A1 = -25/22
A2 = A1 - 1
A3 = A2 - 1
...
A2012 = A2011 - 1
Since A1 is given as -25/22, we can calculate A2012 by repeatedly subtracting 1:
A2012 = (-25/22) - 1 - 1 - ... - 1 (2012 times)
To calculate this, we can multiply -1 by 2012 and divide the result by 22:
A2012 = (-25 - 2012)/22
Simplifying the numerator gives:
-2037/22
So, the final output after 2012 operations is -2037/22.
Therefore, the values of a and b are -2037 and 22, respectively.