Math Problem Statement
What is the mathematical function f(n) = { n/2 if n is even, 3n + 1 if n is odd } and its explanation?
Solution
This function represents the Collatz function , which is commonly associated with the Collatz conjecture. Here's the function breakdown:
\begin{cases} \frac{n}{2} & \text{if } n \text{ is even} \\ 3n + 1 & \text{if } n \text{ is odd} \end{cases}$$ ### Explanation: 1. **If $$n$$ is even**: The function simply divides $$n$$ by 2. For example, if $$n = 8$$, then $$f(8) = 4$$. 2. **If $$n$$ is odd**: The function multiplies $$n$$ by 3 and then adds 1. For example, if $$n = 7$$, then $$f(7) = 3(7) + 1 = 22$$. --- ### Application of the Collatz Function: This function is used iteratively in the Collatz conjecture, which states: - Starting with any positive integer $$n$$, repeatedly applying $$f(n)$$ will eventually reach the number 1. For example, starting with $$n = 6$$: $$6 \to 3 \to 10 \to 5 \to 16 \to 8 \to 4 \to 2 \to 1$$ Would you like me to demonstrate the iterations for a specific $$n$$ or explain the mathematical significance further? --- ### Related Questions: 1. What happens when the function is applied to a specific even or odd number repeatedly? 2. How does the Collatz conjecture relate to number theory? 3. Can we prove the Collatz conjecture for all positive integers? 4. What are the longest sequences known before reaching 1 for specific starting values of $$n$$? 5. How does modular arithmetic help in understanding the function? ### Tip: To explore the Collatz function further, try starting with different values of $$n$$ and trace the steps. Patterns often emerge!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Number Theory
Iterative Functions
Formulas
f(n) = n/2 if n is even
f(n) = 3n + 1 if n is odd
Theorems
Collatz Conjecture
Suitable Grade Level
Grades 10-12
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