Step-by-Step Instructions
Choose a Starting Number
Select any positive integer as your starting number (n). This will be the first number in your Collatz sequence.
Apply the Collatz Formula
If your starting number is even, divide it by 2 to get the next number in the sequence. If it's odd, multiply it by 3 and add 1. Write down the result as the next number in your sequence.
Repeat the Process
Take the new number you obtained in the previous step and apply the Collatz formula again. Continue this process until you reach the number 1, as the sequence will always eventually reach 1 according to the conjecture.
Track the Total Steps and Peak Value
As you generate each number in the sequence, keep track of how many steps it takes to reach 1. Also, note the highest number (peak value) you encounter during this process.
Worked Example
Let's start with the number 6. Since 6 is even, we divide it by 2 to get 3. Then, since 3 is odd, we multiply it by 3 and add 1 to get 10. We continue this process: 10 is even, so we divide by 2 to get 5. Since 5 is odd, we multiply by 3 and add 1 to get 16. Continuing, 16 is even, so we divide by 2 to get 8, then 4, then 2, then 1. The sequence is 6, 3, 10, 5, 16, 8, 4, 2, 1. It took 8 steps to reach 1, and the peak value was 16.
Common Mistakes to Avoid and Using the Calculator
A common mistake is not correctly applying the formula based on whether the number is odd or even. To avoid this, double-check each calculation. For convenience and to avoid errors, especially with large starting numbers, you can use a Collatz sequence calculator. However, understanding how to do it manually will help you appreciate the sequence's properties and behavior.
Introduction to the Collatz Sequence
The Collatz sequence, also known as the 3n+1 conjecture, is a series of numbers produced from a starting number, following two simple rules. In this guide, we will walk you through the steps to calculate the Collatz sequence manually.
What is the Collatz Sequence Formula?
The Collatz sequence is generated using the following formula:
- If the current number is even, the next number is half of the current number (n/2).
- If the current number is odd, the next number is 3 times the current number plus 1 (3n+1).
Step-by-Step Guide
To calculate the Collatz sequence, follow these steps: