Problem 104 - Calculus for Biology and Medicine

Question

The sequence \(\left\\{a_{n} \mid\right.\) is recursively defined. Find all fixed points of \(\left[a_{n}\right\\} .\) $$ a_{n+1}=a_{n}^{2}-a_{n} $$

Step-by-Step Solution

Answer

The fixed points are 0 and 2.

Step 1: Understanding Fixed Points

A fixed point in a sequence is a value such that when you plug it into the recursion formula, the value remains the same, i.e., it doesn't change with each iteration. For the sequence \( \{a_n\} \), if \( a_{n} \) is a fixed point then \( a_{n+1} = a_{n} \). This implies that \( a_{n}^2 - a_{n} = a_{n} \).

Step 2: Setting the Equation for Fixed Points

To find the fixed points, we set up the equation \( a_{n}^2 - a_{n} = a_{n} \). Simplifying this gives \( a_{n}^2 - 2a_{n} = 0 \).

Step 3: Factoring the Equation

The equation \( a_{n}^2 - 2a_{n} = 0 \) can be factored as \( a_{n}(a_{n} - 2) = 0 \). This provides potential solutions for \( a_{n} \).

Step 4: Finding the Fixed Points

From the factored equation \( a_{n}(a_{n} - 2) = 0 \), we find the possible solutions: \( a_{n} = 0 \) and \( a_{n} = 2 \). These are the fixed points of the sequence.

Key Concepts

Recurrence Relations

Recurrence relations are formulas that define each term of a sequence in terms of its previous terms. They play a crucial role in sequences as they allow us to understand how sequences behave over time. In the exercise, the sequence is given by the recurrence relation \( a_{n+1} = a_{n}^2 - a_{n} \). This means every term, \( a_{n+1} \), depends on the term before it, \( a_{n} \), based on the formula.

• Recurrence relations simplify complex sequences into manageable parts.
• They help in analyzing the growth or decay of sequences over iterations.

Recurrence relations are often used in various mathematical, computer science, and engineering problems. They describe systems where the state evolves over time. By observing how \( a_{n} \) transforms into \( a_{n+1} \), you learn about the sequence's trajectory and stability.

Sequence Convergence

Sequence convergence deals with whether or not a sequence approaches a specific value as you go to infinity. In simpler terms, if you continue plotting the sequence, does it settle at a specific number?
For the given sequence defined by the formula \( a_{n+1} = a_{n}^2 - a_{n} \), the fixed points \( a_n = 0 \) and \( a_n = 2 \) indicate points of convergence. These are values that, if reached, remain unchanged through further iterations. In mathematical terms, they are the "fixed" positions where a sequence can settle.

• If a sequence converges, it approaches one of its fixed points.
• Only if the initial condition or the transformation in the sequence aligns with these fixed points, does convergence occur.

Understanding convergence helps in predicting long-term behavior in dynamic systems, such as calculating stable states and equilibrium points.

Factoring Quadratic Equations

Factoring quadratic equations is a method used to solve equations of the form \( ax^2 + bx + c = 0 \). This method was used in the solution to find fixed points of the sequence by factoring the expression \( a_{n}^2 - 2a_{n} = 0 \).
To factor this equation:

• Look for common factors. Here \( a_{n} \) is a common factor in \( a_{n}^2 - 2a_{n} \), resulting in \( a_{n}(a_{n} - 2) \).
• Set each factor to zero: \( a_{n} = 0 \) and \( a_{n} - 2 = 0 \).
• Solve each equation to find the fixed points, which are \( a_{n} = 0 \) and \( a_{n} = 2 \).

Factoring is a powerful tool in algebra, allowing you to simplify and solve quadratic equations quickly. This makes it easier to find crucial points in sequences, like fixed points, that help in understanding the sequence's behavior.