Problem 87 - Precalculus: Mathematics for Calculus

Question

Find the first 40 terms of the sequence defined by $$a_{n+1}=\left\\{\begin{array}{ll}\frac{a_{n}}{2} & \text { if } a_{n} \text { is an even number } \\\3 a_{n}+1 & \text { if } a_{n} \text { is an odd number }\end{array}\right.$$ and \(a_{1}=11 .\) Do the same if \(a_{1}=25 .\) Make a conjecture about this type of sequence. Try several other values for \(a_{1}\) to test your conjecture.

Step-by-Step Solution

Answer

The sequence eventually reaches the cycle 4, 2, 1 for any initial value \(a_1\). This is in line with the Collatz Conjecture.

Step 1: Understanding the Sequence Rule

The sequence is defined by a recurrence relation based on the parity of the term. If the term \(a_n\) is even, the next term \(a_{n+1}\) is \(\frac{a_n}{2}\). If \(a_n\) is odd, the next term is calculated as \(3a_n + 1\). We will compute the terms based on this rule.

Step 2: Compute Terms Starting from \(a_1 = 11\)

Start with \(a_1 = 11\):1. \(a_1 = 11\) (odd), so \(a_2 = 3(11) + 1 = 34\)2. \(a_2 = 34\) (even), so \(a_3 = \frac{34}{2} = 17\)3. \(a_3 = 17\) (odd), so \(a_4 = 3(17) + 1 = 52\)4. \(a_4 = 52\) (even), so \(a_5 = \frac{52}{2} = 26\)5. \(a_5 = 26\) (even), so \(a_6 = \frac{26}{2} = 13\) 6. Continue this process up to 40 terms.

Step 3: Compute Terms Starting from \(a_1 = 25\)

Start with \(a_1 = 25\):1. \(a_1 = 25\) (odd), so \(a_2 = 3(25) +1 = 76\)2. \(a_2 = 76\) (even), so \(a_3 = \frac{76}{2} = 38\)3. \(a_3 = 38\) (even), so \(a_4 = \frac{38}{2} = 19\)4. \(a_4 = 19\) (odd), so \(a_5 = 3(19) + 1 = 58\)5. \(a_5 = 58\) (even), so \(a_6 = \frac{58}{2} = 29\) 6. Continue this process up to 40 terms.

Step 4: Identify Patterns and Make a Conjecture

Observing the computed terms, both sequences fluctuate between large and small values, repeatedly encountering the cycle 4, 2, 1, 4,... when a lower number is reached. A conjecture for this type of sequence, known as the Collatz Conjecture, is that regardless of the starting number \(a_1\), the sequence will eventually reach the cycle 4, 2, 1.

Step 5: Test the Conjecture with Other Values for \(a_1\)

Try different initial values for \(a_1\) such as 7, 15, or 20. Observe that each of these values eventually leads to the recurring cycle 4, 2, 1. This testing supports the conjecture made in the previous step about the sequence's behavior.

Key Concepts

Sequence

A sequence is essentially a list of numbers that follow a certain rule or pattern. In the context of the Collatz Conjecture, we look at numbers generated by specific rules based on their previous value. This particular exercise involves generating sequences using an interesting rule:

• If a number (or term) in the sequence is even, divide it by two to get the next term.
• If a number is odd, multiply it by three and add one to get the next term.

Starting from a given number, such as 11 or 25, these rules are applied repeatedly to generate a series of terms. Understanding how sequences are structured and generated is crucial in mathematics since they are a foundational concept that allows us to analyze and identify patterns or properties within sets of numbers.

Recurrence Relation

A recurrence relation defines each term of a sequence using the preceding term(s). It's like a formula that tells you how to get from one term to the next. In the Collatz Conjecture, the recurrence relation uses parity to decide the next move:

• For an even number: \( a_{n+1} = \frac{a_n}{2} \)
• For an odd number: \( a_{n+1} = 3a_n + 1 \)

This kind of relation is very powerful. It provides a simple set of instructions for generating complex patterns. Although the rules here are simple, the behavior of the sequence can be surprisingly intricate, leading to provocative questions and hypotheses like the Collatz Conjecture itself.

Parity

Parity is a concept related to numbers being even or odd. It's a simple yet crucial property that significantly influences the behavior of the sequence in the Collatz Conjecture. In mathematics, parity determines whether a number can be divided evenly by two:

• Even Numbers: These numbers divide by two without a remainder. In the sequence, whenever you encounter an even number, you simply divide by 2 for the next term.
• Odd Numbers: These numbers do not divide evenly by two. For odd numbers, the sequence instructs you to multiply by three and add one.

Recognizing whether a number is even or odd helps you apply the correct part of the recurrence relation, effectively driving the sequence forward toward its eventual outcome.

Pattern Recognition

Pattern recognition is the ability to observe and identify regularities and repeated motifs within data. In exploring sequences generated by the Collatz Conjecture, pattern recognition plays a key role in forming hypotheses and conjectures about the sequence's behavior. As you compute the sequence beginning with any initial value, you will often discover an intriguing and recurring cycle: 4, 2, 1. This cycle suggests that no matter what starting number you choose, the sequence will eventually converge to this loop. Recognizing such patterns can help in posing generalizations and conjectures about broader classes of sequences. Indeed, the Collatz Conjecture itself is born out of noticing this consistent behavior across many different starting values. The fundamental task of mathematics often involves recognizing such patterns and using them to predict and understand more about the systems being studied.