Question

HELP FOR 25 POINTS!

Function $C$ is defined on positive integers as follows:

\[C(n) = \begin{cases}

\dfrac n 2 & \text{if $n$ is even}, \\

3n+1 & \text{if $n$ is odd}.

\end{cases}\]Find the smallest positive integer $m$ such that $C^{m}(9) = 1$. What does this notation mean?

Answer 1

The smallest positive integer m such that C^m(9) = 1 is found by applying the function C to the number 9 repeatedly until reaching 1. After performing the appropriate operations as defined by the function for even and odd integers, the smallest m is determined to be 19.

Step-by-step explanation:

The function C(n) given in the question is a step function defined differently for even and odd positive integers, often associated with the Collatz conjecture in mathematics. The notation Cm(9) means applying the function C to the number 9, m times repeatedly. To find the smallest positive integer m such that Cm(9) = 1, we need to apply the function C to 9 and keep track of the number of applications until we reach 1.

Starting with n = 9 (which is odd):

1. C(9) = 3(9) + 1 = 28 (Even, so divide by 2)
2. C(28) = 28 / 2 = 14 (Even, so divide by 2)
3. C(14) = 14 / 2 = 7 (Odd, so apply 3n + 1)
4. Continue this pattern until the sequence reaches 1.

The smallest m is the count of such applications. By following through the computation, we find that m is 19.