Problem 109 - Calculus for Biology and Medicine

Question

The sequence \(\left\\{a_{n} \mid\right.\) is recursively defined. Find all fixed points of \(\left[a_{n}\right\\} .\) $$ a_{n+1}=\sqrt{5 a_{n}} $$

Step-by-Step Solution

Answer

The fixed points are 0 and 5.

Step 1: Understanding Fixed Points

A fixed point for a sequence or function is a value that remains unchanged when applied to the function. For the given sequence, we are looking for a value of \(a_n\) such that it is equal to \(a_{n+1}\). This means we need to solve the equation \(a_n = \sqrt{5a_n}\).

Step 2: Setting Up the Equation

To find the fixed points, set \(a_n = \sqrt{5a_n}\). This will give us an equation we can solve to find all values of \(a_n\) that satisfy this condition.

Step 3: Solve the Equation

Start by squaring both sides of the equation to eliminate the square root: \[ a_n^2 = 5a_n \] Rearrange the equation: \[ a_n^2 - 5a_n = 0 \] Factor the quadratic equation: \[ a_n(a_n - 5) = 0 \] This gives us two possible solutions: either \(a_n = 0\) or \(a_n = 5\).

Step 4: Validate the Fixed Points

Check if these solutions are indeed fixed points. For \(a_n = 0\): The condition becomes \(a_{n+1} = \sqrt{5 imes 0} = 0\). For \(a_n = 5\): The condition becomes \(a_{n+1} = \sqrt{5 imes 5} = \sqrt{25} = 5\). Both conditions hold, so \(a_n = 0\) and \(a_n = 5\) are indeed fixed points.

Key Concepts

Recursive Sequences

A recursive sequence is a sequence in which each term is defined using one or more of the previous terms. This type of sequence gives us a powerful way to create functions and model behaviors where each step or term depends on the preceding ones. In our specific example, a recursive sequence is expressed with the formula \( a_{n+1} = \sqrt{5a_n} \), where each term \( a_{n+1} \) depends on the term before it, \( a_n \). There are important aspects to understand in recursive sequences:

• **Initial Condition**: To generate a recursive sequence, typically an initial term (or terms) must be given, which becomes the basis for generating the rest of the sequence.
• **Recursive Formula**: The rule or formula, which in our case is \( a_{n+1} = \sqrt{5a_n} \), describes how to calculate the next term based on previous terms.
• **Fixed Points**: These are special values within a sequence that remain constant when applied in the recursive formula, leading to a steady state.

Understanding recursive sequences helps in various domains such as computer algorithms, financial predictions, and even in natural phenomena modeling. The idea is to observe how patterns evolve over iterations.

Quadratic Equations

A quadratic equation is a second-degree polynomial, generally written as \( ax^2 + bx + c = 0 \). This type of equation is essential in many mathematical problems because they often appear when modeling real-world phenomena or solving problems involving squared terms. In our exercise, after applying the recursive formula, we derive a quadratic equation: \[ a_n^2 - 5a_n = 0 \]Here's how we approach solving it:

• **Factoring**: The expression is factored to \( a_n(a_n - 5) = 0 \), providing two potential solutions. This method involves expressing the quadratic in terms of two products equaling zero.
• **Zero Product Property**: Once factored, each factor is set to zero. This property states that if a product of factors equals zero, at least one of the factors must be zero. Hence, we find the solutions \( a_n = 0 \) or \( a_n = 5 \).

Quadratic equations are a fundamental concept not only in solving sequences but also in understanding parabola characteristics, physics trajectories, and optimization problems.

Square Roots

The square root operation is finding a number which, when multiplied by itself, gives the original number. It's commonly represented using the radical symbol \( \sqrt{} \). In this exercise, we're dealing with square roots through the recursive formula \( a_{n+1} = \sqrt{5a_n} \).Understanding square roots involves:

• **Principal Square Root**: This is the non-negative square root of a number. For example, \( \sqrt{16} = 4 \), because 4 is non-negative and \( 4^2 = 16 \).
• **Rational and Irrational Roots**: The results can be either rational (like \( \sqrt{25} = 5 \)) or irrational (like \( \sqrt{2} \)), depending on whether the original number is a perfect square.
• **Application in Functions**: In our fixed points problem, square roots are used to transform the sequence, impacting \( a_{n+1} \) directly. Manipulating equations involving square roots often requires caution such as squaring both sides to simplify, as done when solving for fixed points.

Square roots are critical in many areas of math from solving equations to geometry and calculus, offering means to understand proportions and magnitudes across various contexts.