Problem 103 - Calculus for Biology and Medicine

Question

In Problems 103-110, assume that \(\lim _{n \rightarrow \infty} a_{n}\) exists. Find all fixed ooints of \(\left\\{a_{n}\right\\}\), and use a table or other reasoning to guess which ixed point is the limiting value for the given initial condition. $$ a_{n+1}=\frac{1}{2}\left(a_{n}+5\right), a_{0}=1 $$

Step-by-Step Solution

Answer

The fixed point is 5, and the sequence converges to this value.

Step 1: Define a Fixed Point

A fixed point for the sequence is a value \( x \) such that \( a_{n+1} = a_n = x \). This means that substituting this into the given relation for the sequence must hold true for the fixed point.

Step 2: Set Up the Fixed Point Equation

Substitute \( a_{n+1} = a_n = x \) into the equation \( a_{n+1} = \frac{1}{2}(a_n + 5) \). This gives the equation \( x = \frac{1}{2}(x + 5) \).

Step 3: Solve the Fixed Point Equation

Multiply both sides by 2 to get rid of the fraction: \( 2x = x + 5 \). Then solve for \( x \) by subtracting \( x \) from both sides, yielding \( x = 5 \). Thus, the fixed point is \( 5 \).

Step 4: Initialize the Sequence and Calculate Terms

With the initial condition \( a_0 = 1 \), compute the first few terms of the sequence to form an educated guess regarding its behavior. Using the formula: \( a_1 = \frac{1}{2}(1 + 5) = 3 \), \( a_2 = \frac{1}{2}(3 + 5) = 4 \), \( a_3 = \frac{1}{2}(4 + 5) = 4.5 \).

Step 5: Observe the Limiting Behavior

The sequence \( \{a_n\} \) appears to be approaching the fixed point \( 5 \) as more terms are computed. This is a strong indicator that the limiting value of the sequence as \( n \to \infty \) is indeed \( 5 \).

Key Concepts

Sequence Convergence

In mathematics, when we talk about sequences, a significant concept is sequence convergence. This describes the behavior of a sequence as it progresses indefinitely. When a sequence converges, it approaches a specific value, known as the limit, as more and more terms are added. If we consider the sequence given in our exercise, the convergence idea should clarify a lot of questions.
For example, in our problem, we saw the sequence generated by the recurrence relation \( a_{n+1} = \frac{1}{2}(a_n + 5) \) with an initial value of \( a_0 = 1 \). As we calculate further terms like \( a_1 = 3 \), \( a_2 = 4 \), and \( a_3 = 4.5 \), it's evident that these terms are settling closer to a specific value. This trend indicates convergence as \( n \) becomes very large.

• The sequence converges towards the fixed point.
• This fixed point acts as the sequence's limit value.
• Convergence is a key property to know whether the sequence stabilizes or not.

Limit of a Sequence

The limit of a sequence is the value that the terms of a sequence approach as the index (in this case, \( n \)) increases without bound. When the sequence's terms get closer and closer to a certain value, and stay arbitrarily close to this value for indices greater than some number, we say the sequence has a limit.
In the problem at hand, the limit of the sequence is the number that \( a_n \) approaches as \( n \to \infty \). By observing our sequence, generated by the recurrence relation, we noticed its terms converging progressively towards 5. Thus, the limit of this sequence is 5.

• Limits are essential for determining the behavior of sequences at infinity.
• A sequence may either have a finite limit or diverge (not settle on any value).
• Finding the limit helps to understand the long-term behavior of a sequence.

Recurrence Relation

Recurrence relations define sequences using preceding terms. They describe how each term in the sequence relates to its predecessor, thereby creating a predictable pattern that governs the sequence's behavior.
In this specific problem, the recurrence relation \( a_{n+1} = \frac{1}{2}(a_n + 5) \) provides a step-by-step rule for generating each subsequent term based on the previous one. Starting with an initial condition \( a_0 = 1 \), we apply the relation repeatedly to advance further in the sequence.

• Recurrence relations are crucial for specifying sequences succinctly.
• These relations can help predict future terms, examining trends such as convergence.
• Understanding the relation gives insight into the sequence's long-term behavior.

By mastering the use of recurrence relations, you can unravel the secrets of infinite sequences and decode their eventual behavior.