Question

Consider the following sequences defined by a recurrence relation. Use a calculator, analytical methods, and/or graphing to make a conjecture about the limit of the sequence or state that the sequence diverges. $$a_{n+1}=4 a_{n}\left(1-a_{n}\right) ; a_{0}=0.5$$

Short answer

Answer: The limit of the given recurrence relation is 0.

Step-by-step solution

Analytical Method

To analyze the given recurrence relation analytically, let's assume the sequence converges to some limit L, then for large values of n: $$a_{n+1} = L = 4a_{n}(1-a_{n})$$ Now, plugging \(a_{n}=L\) in the equation, we get: $$L = 4L(1-L)$$ Let's find the possible values for L.

Solving the Recursive Relation

To solve the equation, we can simplify it: $$L - 4L^2 = 0$$ Factoring out the L, we get: $$L(1-4L) = 0$$ Thus we have two possible values for L: $$L_1 = 0 \text{ and } L_2 = \frac{1}{4}$$

Calculate a few terms of the sequence

Now let's calculate the first few terms of the sequence using the given initial value \(a_0 = 0.5\). Using the formula and iterating: $$a_1 = 4(0.5)(1 - 0.5) = 1$$ $$a_2 = 4(1)(1 - 1) = 0$$ $$a_3 = 4(0)(1 - 0) = 0$$ $$a_4 = 4(0)(1 - 0) = 0$$ As we can see, once the sequence reaches the term \(a_2=0\), it remains constant at 0, which is also one of our calculated possible limit values \(L_1\).

Conclusion

We can now make a conjecture about the limit of the given sequence. Since the sequence converges to 0, which is one of our possible limit values, we can conclude that the sequence converges and its limit is: $$\lim_{n\to\infty} a_n = 0$$

Key concepts

Convergence of Sequences

Convergence of sequences is a fundamental concept in mathematics. It describes how a sequence behaves as the number of terms increases. When we say a sequence converges, it means that as we continue to calculate more terms (as n approaches infinity), the sequence tends to get closer and closer to a specific number, called the limit.

In the context of the exercise given, the sequence under investigation was generated by a recurrence relation. A recurrence relation uses the previous term(s) to determine the next one. For our sequence:

• The first term given is \(a_0 = 0.5\).
• Each following term is calculated based on the formula \(a_{n+1} = 4a_n(1-a_n)\).

We calculated several terms of the sequence to detect a pattern. When we reached \(a_2=0\), all subsequent terms remained at 0, suggesting that the sequence converges to this value. This determination was consistent with the analytical method we applied, affirming convergence to the limit of 0.

Limit of a Sequence

Understanding the limit of a sequence is critical for analyzing sequences. The limit of a sequence is the value that the terms of the sequence approach as the number of terms goes to infinity. If the terms get closer and closer to a particular value, we identify this as the limit.

For the sequence from the exercise:

• We examined the equation derived from assuming convergence: \( L = 4L(1-L) \).
• Solving this equation revealed potential limit values: \( L = 0 \) or \( L = \frac{1}{4} \).

By computing the sequence, we saw it stabilized at 0 after \(a_2\). This confirmed our theory that 0 is the actual limit. Recognizing the correct limit helps us understand the ultimate behavior of the sequence. In practical terms, knowing the limit can help predict long-term outcomes without needing to calculate endless terms.

Analytical Methods

Utilizing analytical methods is essential for solving recurrence relations systematically. Unlike numerical approximation, analytical methods provide an exact solution by manipulating the algebraic formula or equation involved.

To find the limit of the sequence in the exercise, we assumed convergence and set \( a_n \) equal to \( L \), the hypothetical limit. This approach allowed us to derive an equation reflecting the steady state of the sequence:

• We started with \( L = 4L(1-L) \), a common step when analyzing limits of recursive sequences.
• This equation was simplified to \( L(1-4L) = 0 \), which we could solve by factoring.

Finding solutions of \( L = 0 \) and \( L = \frac{1}{4} \), we postulated potential sequence limits. With further verification by calculating more terms, we established the correct limit of the sequence effectively. Analytical methods provide the tools necessary to break down complex problems into understandable parts, making them incredibly valuable for understanding sequences and their behaviors.