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?
Answers
Reiny
A<sub>1</sub> = 22/47
A<sub>2</sub> = (22/47 - 1)/(22/47
= -25/22
A<sub>3</sub> = (-25/22 - 1)/(-25/22)
= 47/25
A<sub>4</sub> = (47/25 - 1)/(47/25)
= 22/47
A<sub>5</sub> = -25/22
A<sub>6</sub> = 47/25
A<sub>7</sub> = 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
A<sub>2</sub> = (22/47 - 1)/(22/47
= -25/22
A<sub>3</sub> = (-25/22 - 1)/(-25/22)
= 47/25
A<sub>4</sub> = (47/25 - 1)/(47/25)
= 22/47
A<sub>5</sub> = -25/22
A<sub>6</sub> = 47/25
A<sub>7</sub> = 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
Steve
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, f<sup>2012</sup>(A) = f<sup>2</sup>(A)
so, take a look:
f(22/47) = 1 - 47/22 = -25/22
f(-25/22) = 1 + 22/25 = 47/25
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, f<sup>2012</sup>(A) = f<sup>2</sup>(A)
so, take a look:
f(22/47) = 1 - 47/22 = -25/22
f(-25/22) = 1 + 22/25 = 47/25
Reiny
Steve is right, it is a cycle of 3 terms, not 4 terms like I hastily concluded.
so -25/22
so -25/22