Problem 93
Question
In Problems \(93-102\), the sequence \(\left\\{a_{n}\right\\}\) is recursively defined. Find all fixed points of \(\left\\{a_{n}\right\\}\) $$ a_{n+1}=\frac{1}{2} a_{n}+2 $$
Step-by-Step Solution
Verified Answer
The fixed point is 4.
1Step 1: Understand the Concept of a Fixed Point
A fixed point in a sequence is a value \(x\) such that when passed through the sequence's function, it maps to itself. For the given recursive relation, \(a_{n+1} = \frac{1}{2}a_n + 2\), we want to find a value of \(a_n = x\) that satisfies \(a_{n+1} = a_n\).
2Step 2: Set Up the Fixed Point Equation
Since a fixed point is a value that doesn't change as the sequence progresses, set \(a_{n+1} = a_n = x\) in the recursive equation. This gives us the equation: \(x = \frac{1}{2}x + 2\).
3Step 3: Solve the Fixed Point Equation
Solve the equation \(x = \frac{1}{2}x + 2\) by eliminating the fraction. Subtract \(\frac{1}{2}x\) from both sides to isolate \(x\): \[x - \frac{1}{2}x = 2\].
4Step 4: Simplify the Equation
Simplify the left-hand side: \(\frac{1}{2}x = 2\). Multiply both sides by 2 to solve for \(x\): \(x = 4\).
5Step 5: Verification
Verify the solution by checking if substituting \(x = 4\) into the original sequence function results in \(4\). Substituting gives: \[a_{n+1} = \frac{1}{2}(4) + 2 = 2 + 2 = 4\], which equals \(x\). Thus, \(x = 4\) is a confirmed fixed point.
Key Concepts
Recursive SequencesMathematical SequencesCalculus Concepts
Recursive Sequences
A recursive sequence is defined by a rule that computes each term using one or more of the preceding terms. This means you do not simply apply a formula to find the terms but rather use the relationship between terms to build up the sequence.
In the context of our problem, the sequence \(a_{n+1} = \frac{1}{2}a_n + 2\) is defined recursively. Here, each term \(a_{n+1}\) depends on the previous term \(a_n\). This type of sequence can often be easier to compute iteratively, going step-by-step from a starting value. This recursive method can help solve real-world problems, as it simulates processes that evolve step by step, such as populations or financial balances.
Key points about recursive sequences:
In the context of our problem, the sequence \(a_{n+1} = \frac{1}{2}a_n + 2\) is defined recursively. Here, each term \(a_{n+1}\) depends on the previous term \(a_n\). This type of sequence can often be easier to compute iteratively, going step-by-step from a starting value. This recursive method can help solve real-world problems, as it simulates processes that evolve step by step, such as populations or financial balances.
Key points about recursive sequences:
- Each term is calculated using previous terms.
- You need a starting value, often called the base case.
- Recursive formulas can model dynamic processes.
Mathematical Sequences
Mathematical sequences are ordered lists of numbers that often follow a specific rule or pattern. There are various types of sequences, such as arithmetic, geometric, and recursive sequences, each with their own defining characteristics. In our exercise, \(a_{n+1} = \frac{1}{2}a_n + 2\) is a recursive sequence that generates a list of numbers based on each previous number.
For this kind of sequence, understanding fixed points is vital. A fixed point occurs when a sequence number remains constant through the sequence’s rule, essentially \(a_n = a_{n+1}\). When a sequence reaches a fixed point, it stops changing, no matter how many iterations are performed.
Useful aspects of mathematical sequences:
For this kind of sequence, understanding fixed points is vital. A fixed point occurs when a sequence number remains constant through the sequence’s rule, essentially \(a_n = a_{n+1}\). When a sequence reaches a fixed point, it stops changing, no matter how many iterations are performed.
Useful aspects of mathematical sequences:
- Helps identify trends and patterns.
- Provides insight into stability and convergence of sequences.
- Applies to real-world situations like signal processing or economic models.
Calculus Concepts
Calculus concepts are often useful in analyzing and understanding sequences, especially when looking for features like fixed points. In this context, fixed points particularly relate to ideas about limits and convergence, where sequences stabilize as terms progress. Calculus provides the tools to analyze such behavior by investigating limits.
Here’s how it connects:
Here’s how it connects:
- A fixed point is similar to a limit where subsequent terms stabilize to a single value.
- Analyzing sequence behavior involves exploring continuity and change, core ideas in calculus.
- Through calculus, we can determine how a sequence behaves as it approaches a fixed point.
Other exercises in this chapter
Problem 81
Use the limit laws to determine \(\lim _{n \rightarrow \infty} a_{n}=a .\) $$ \lim _{n \rightarrow \infty} \frac{n+2^{-n}}{n} $$
View solution Problem 82
Use the limit laws to determine \(\lim _{n \rightarrow \infty} a_{n}=a .\) $$ \lim _{n \rightarrow \infty} \frac{n+3^{-n}}{n} $$
View solution Problem 94
The sequence \(\left\\{a_{n}\right\\}\) is recursively defined. Find all fixed points of \(\left\\{a_{n}\right\\}\) $$ a_{n+1}=\frac{1}{3} a_{n}+\frac{4}{3} $$
View solution Problem 95
The sequence \(\left\\{a_{n}\right\\}\) is recursively defined. Find all fixed points of \(\left\\{a_{n}\right\\}\) $$ a_{n+1}=\frac{2}{5} a_{n}-\frac{9}{5} $$
View solution