An operation produces A−1/A from a fraction A= m/n, where m ≠ n and m ≠ 0. If the initial value of A is 24/47 and the operation is repeated 2012 times, the final output is a/b, where a and b are coprime positive integers. What is a+b?

To find the final output of the operation after it is repeated multiple times, we need to perform the operation repeatedly on the initial value.

Let's start by performing the operation on the initial value A = 24/47:
(A - 1) / A = (24/47 - 1) / (24/47)

To simplify the expression, we first find a common denominator:
= (24/47 - 47/47) / (24/47)
= (-23/47) / (24/47)

When dividing by a fraction, it is the same as multiplying by its reciprocal:
= (-23/47) * (47/24)

Now, let's continue performing the operation on the result:
((-23/47) * (47/24) - 1) / ((-23/47) * (47/24))

Again, simplifying the expression:
= ((-23/24) - 1) / ((-23/24))

= (-23/24 - 24/24) / (-23/24)
= (-47/24) / (-23/24)

= (-47/24) * (24/23)
= -47/23

So, after 1 operation, the output is -47/23.

Now, let's repeat the operation 2012 times. Since the operation does not depend on the numerator or denominator values, the output will remain the same after each operation.

Therefore, after 2012 operations, the final output will still be -47/23.

Since we need to express the final output as a fraction in the form a/b with coprime positive integers, we can multiply both the numerator and denominator by -1 to make them positive:

(-47) * (-1) / (23) * (-1) = 47/23

So, the final output after 2012 operations is 47/23.

The values of a and b are 47 and 23, respectively.

Therefore, a + b = 47 + 23 = 70.

Hence, the value of a + b is 70.