Problem 96 - Calculus for Biology and Medicine

Question

The sequence \(\left\\{a_{n}\right\\}\) is recursively defined. Find all fixed points of \(\left\\{a_{n}\right\\}\) $$ a_{n+1}=-\frac{1}{3} a_{n}+\frac{1}{4} $$

Step-by-Step Solution

Answer

The fixed point of the sequence is \(a_n = \frac{3}{16}\).

Step 1: Understand the Problem

We are tasked with finding the fixed points of a sequence \(\{a_{n}\}\) that is recursively defined by the formula \(a_{n+1} = -\frac{1}{3} a_n + \frac{1}{4}\). A fixed point is a value of \(a_n\) such that it remains unchanged in the next iteration, i.e., \(a_{n+1} = a_n\).

Step 2: Set Up the Fixed Point Equation

To find the fixed points, set \(a_{n+1} = a_n\) in the recursive equation. Thus, the equation becomes \(a_n = -\frac{1}{3} a_n + \frac{1}{4}\).

Step 3: Solve for Fixed Points

Rearrange and solve the equation to find the value of \(a_n\). Start with: \(a_n = -\frac{1}{3} a_n + \frac{1}{4}\). Add \(\frac{1}{3} a_n\) to both sides to obtain \(a_n + \frac{1}{3} a_n = \frac{1}{4}\), which simplifies to \(\frac{4}{3} a_n = \frac{1}{4}\). Multiply both sides by \(\frac{3}{4}\) to solve for \(a_n\), giving \(a_n = \frac{3}{16}\).

Step 4: Verify the Solution

Substitute \(a_n = \frac{3}{16}\) back into the original recursive equation to ensure it holds: Calculate \(-\frac{1}{3} \cdot \frac{3}{16} + \frac{1}{4}\). Simplify \(-\frac{1}{3} \cdot \frac{3}{16}\) to \(-\frac{1}{16}\). Adding \(\frac{1}{4}\) (which is \(\frac{4}{16}\)) gives \(\frac{3}{16}\). The result matches the fixed point we found, confirming its correctness.

Key Concepts

Fixed Points of a Recursive Sequence

In the context of sequences, a fixed point is a specific value within the sequence that remains consistent across iterations. For a sequence defined by the recurrence relation \(a_{n+1} = -\frac{1}{3} a_n + \frac{1}{4}\), finding a fixed point requires finding a value of \(a_n\) such that \(a_{n+1} = a_n\). This means the value does not change as you move from one term to the next.
To determine if a fixed point exists, set up the equation by equating \(a_{n+1}\) to \(a_{n}\). From the recurrence relation, this becomes \(a_n = -\frac{1}{3} a_n + \frac{1}{4}\). Once set up, we solve the equation for \(a_n\).

• Rearrange the equation to consolidate terms involving \(a_n\).
• Simplify and solve to isolate \(a_n\) on one side.
• The solution \(a_n = \frac{3}{16}\) is our fixed point, meeting the requirement \(a_{n+1} = a_n\).

By understanding fixed points, you can predict elements within a sequence that stabilize, providing a constant value throughout the series of calculations.

Understanding Recurrence Relations

A recurrence relation gives a method for defining sequences based on previous terms. These are rules that determine the next term in a sequence using one or more of the preceding terms. The sequence \(\{a_n\}\) is defined by the relation \(a_{n+1} = -\frac{1}{3} a_n + \frac{1}{4}\), indicating how each term relates to its predecessor.

• Recurrence relations can range from linear to non-linear, simple to complex.
• They can be used to model behaviors over time, such as population growth or financial calculations.

When approaching a recurrence relation, it is important to:

• Note the coefficients and constants, as these determine the relationships' strength and direction.
• Pay attention to signs; for example, a negative coefficient suggests a decrease in value across terms.

Through recurrence relations, the evolution of sequences can be systematically computed, revealing patterns and stable points like the fixed points described earlier.

Sequence Analysis and Its Importance

Sequence analysis involves examining sequences to identify patterns, convergence, or other notable behaviors. By analyzing the given sequence defined by \(a_{n+1} = -\frac{1}{3} a_n + \frac{1}{4}\), we unravel the behavior of \(\{a_n\}\). This entails:

• Identifying the sequence behavior: Are there values or limits where the sequence settles?
• Finding fixed points to understand stability.
• Looking for periodicity or other recurring patterns.

Analyzing sequences aids in understanding how values change and stabilize over iterations. It provides insights into systems modeled over time, allowing predictions and interpretations of long-term behavior.
For instance, if we discover that \(a_n\) converges to a value as \(n\) approaches infinity, this provides conclusions about the series' behavior in different scenarios. Sequence analysis extends beyond mathematics to various fields, where understanding underlying patterns is crucial for decision-making and forecast development.