91Ó°ÊÓ

Let \(a_n\) be the number of different strings of length \(n\) that can be formed from the symbols \(\mathrm{X}\) and \(\mathrm{O}\) with the restriction that a string may not consist of identical smaller strings. For example, XXXX and XOXO are not allowed. The possible strings of length 4 are XXXO, XXOX, XXOO, XOXX, XOOX, XOOO, OXXX, OXXO, OXOO, OOXX, OOXO, OOOX, so \(a_4=12\). Here is a table showing \(a_n\) and the ratio \(\frac{a_{n+1}}{a_n}\) for \(n=1,2, \ldots, 13\). $$ \begin{array}{cccccccccccccc} n & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 \\ \hline a_n & 2 & 2 & 6 & 12 & 30 & 54 & 126 & 240 & 504 & 990 & 2046 & 4020 & 8190 \\ \frac{a_{n+1}}{a_n} & 1 & 3 & 2 & 2.5 & 1.8 & 2.33 & 1.90 & 2.1 & 1.96 & 2.07 & 1.96 & 2.03 & 1.98 \end{array} $$ The table suggests two conjectures: a. For any \(n>2, a_n\) is divisible by 6 . b. \(\lim _{n \rightarrow \infty} \frac{a_{n+1}}{a_n}=2\). Prove or disprove each of these conjectures.

Step by step solution

01

Verify divisibility by 6

First, check the given values of \(a_n\) to see if they are divisible by 6 for \(n > 2\). From the table, the values of \(a_n\) for \(n > 2\) are 6, 12, 30, 54, 126, 240, 504, 990, 2046, 4020, and 8190. Check the divisibility of these by 6. Notice that all these numbers are divisible by 6.

02

Induction proof for divisibility by 6

To generalize, use induction to prove that \(a_n\) is divisible by 6 for \(n > 2\). Base case: For \(n = 3\), \(a_3 = 6\), which is divisible by 6. Inductive step: Assume \(a_k\) is divisible by 6 for some \(k \geq 3\), i.e., \(a_k = 6m\) for some integer \(m\). Show that \(a_{k+1}\) is also divisible by 6 by using known recursion or properties of the sequence, which often involves recurrence relations or counting arguments respecting given constraints.

03

Limit of the ratio

Examine the ratio \(\frac{a_{n+1}}{a_n}\) for larger values of \(n\). The table shows these ratios: 1, 3, 2, 2.5, 1.8, 2.33, 1.9, 2.1, 1.96, 2.07, 1.96, 2.03, and 1.98. As \(n\) increases, these ratios seem to approach 2. Use the sequence's properties and limiting behaviors to rigorously show that \(\lim_{n \to \infty} \frac{a_{n+1}}{a_n} = 2\).

04

Proof using recurrence relation

If a recurrence relation for \(a_n\) can be identified and solved, use it to show that the limit of \(\frac{a_{n+1}}{a_n}\) as \(n\) approaches infinity is 2. This involves solving or approximating the recursion to observe the long-term behavior of \(a_n\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Combinatorics

Combinatorics is a field of mathematics that studies the counting, arrangement, and combination of objects. In this exercise, we analyze the number of different strings of length \(n\) that can be composed of symbols \( \mathrm{X} \) and \( \mathrm{O} \) with specific restrictions.
Here, we are particularly interested in strings that do not repeat identical smaller strings within them. For example, seeing 'XXXX' or 'XOXO' within a longer string means those combinations are not allowed. Understanding these combinations involves:

• Enumerating possibilities: Listing out all possible strings that fit the criteria.
• Applying restrictions: Disqualifying strings that repeat identical patterns.
• Generating functions: Encapsulating the combinatorial rules into mathematical formulas or functions.

Combinatorics helps us set the stage for recognizing patterns and proving conjectures through structured and systematic counting methods.

Mathematical Induction

Mathematical induction is a powerful proof technique used to establish the validity of a statement for all natural numbers. It's often used in sequence and series analysis to prove properties that hold for all terms. Here's the step-by-step process:

• Base Case: Prove the statement for the first term (or the first few terms) in the sequence.
• Inductive Step: Assume the statement is true for some arbitrary number \(k\). This is known as the inductive hypothesis.
• Inductive Conclusion: Prove that if the statement holds for \(k\), it must also hold for \(k+1\).

In this exercise, we use induction to show that \(a_n\) is divisible by 6 for \(n \gt 2\). We start by checking initial terms and then assume it's true for some term \(k\). Using properties of the sequence and possibly a recurrence relation, we prove it must be true for \(k+1\). This completes our induction process, ensuring the pattern holds indefinitely.

Limits and Continuity

Limits and continuity are fundamental concepts in calculus that help us understand the behavior of sequences as they approach infinity. In this problem, we are interested in the limit of the ratio \( \frac{a_{n+1}}{a_n} \) as \(n\) approaches infinity.
To find this limit, we observe the trend of the ratios from the given table. As \(n\) increases, the ratio \( \frac{a_{n+1}}{a_n} \) seems to approach 2. The formal way to analyze this is by:

• Identifying a pattern: Recognize how the ratios are changing as \(n\) grows.
• Using recurrence relations: Establish if there's a recurrence relation that simplifies the ratio's behavior.
• Applying mathematical limits: Utilize limit theorems to rigorously prove that \( \lim_{n \rightarrow \infty} \frac{a_{n+1}}{a_n} = 2 \).

Understanding limits helps confirm conjectures about long-term behaviors of sequences, ensuring that our observations are consistently true.