Problem 101
Question
In Problems 101-110, the sequence \(\left\\{a_{n} \mid\right.\) 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 of the sequence is 4.
1Step 1: Understand the Fixed Point
The fixed point of a sequence occurs when the value does not change between terms, which means it satisfies the equation: \( a_{n+1} = a_n \).
2Step 2: Set Up the Fixed Point Equation
Using the recursive definition \( a_{n+1} = \frac{1}{2} a_n + 2 \), substitute \( a_{n+1} = a_n \) resulting in the equation \( a_n = \frac{1}{2}a_n + 2 \).
3Step 3: Solve for the Fixed Point
Simplify the equation \( a_n = \frac{1}{2}a_n + 2 \). Subtract \( \frac{1}{2}a_n \) from both sides to get \( \frac{1}{2} a_n = 2 \).
4Step 4: Isolate the Variable
Solve for \( a_n \) by multiplying both sides by 2: \( a_n = 4 \).
Key Concepts
Fixed PointSequence ConvergenceRecursive Definition
Fixed Point
In the realm of sequences, a fixed point is a special value where the sequence stops changing. Think of it as a balancing point where if you start with this value in the sequence, it stays the same in every step.
To find a fixed point, set the equation \(a_{n+1} = a_n\) with the recursive formula provided. Solve for \(a_n\) like any algebraic equation. For instance, in the equation \(a_{n+1} = \frac{1}{2}a_n + 2\), you solve by setting \(a_n = \frac{1}{2}a_n + 2\). The calculations follow through to show that \(a_n\) equals 4, making 4 the fixed point of this sequence.
- A fixed point occurs when the output of a recurring process equals the input.
- In mathematical terms, for a sequence defined as \(a_{n+1} = f(a_n)\), the fixed point satisfies \(a_{n+1} = a_n\).
To find a fixed point, set the equation \(a_{n+1} = a_n\) with the recursive formula provided. Solve for \(a_n\) like any algebraic equation. For instance, in the equation \(a_{n+1} = \frac{1}{2}a_n + 2\), you solve by setting \(a_n = \frac{1}{2}a_n + 2\). The calculations follow through to show that \(a_n\) equals 4, making 4 the fixed point of this sequence.
Sequence Convergence
Sequence convergence is when the terms of a sequence get closer and closer to a specific value as you move along the sequence. This value, if it exists, is called the limit.
In the given recursive sequence \(a_{n+1} = \frac{1}{2}a_n + 2\), our task is to find if it converges. As we found earlier, the fixed point is 4. If we start with some initial value, the sequence values should stabilize and get closer to 4 as you continue through the sequence. This demonstrates that the sequence indeed converges to the fixed point.
- For a sequence defined by a recursive formula, convergence occurs when the terms approach the fixed point.
- A sequence is convergent if it eventually stays arbitrarily close to some limit point, as the number of steps (n) goes to infinity.
In the given recursive sequence \(a_{n+1} = \frac{1}{2}a_n + 2\), our task is to find if it converges. As we found earlier, the fixed point is 4. If we start with some initial value, the sequence values should stabilize and get closer to 4 as you continue through the sequence. This demonstrates that the sequence indeed converges to the fixed point.
Recursive Definition
A recursive definition outlines each term of a sequence in terms of the preceding term. It's like a blueprint guiding how to build the sequence from one step to the next.
For the sequence provided, \(a_{n+1} = \frac{1}{2}a_n + 2\), the formula tells us what the next sequence term will be using the current term \(a_n\). Starting with an initial term, like \(a_0\), you can calculate \(a_1\), then \(a_2\), and so on. Recursive sequences often reveal powerful patterns of growth or decay, depending on how the recursive formula is constructed.
- Each term in the sequence is generated using a formula that involves the preceding terms.
- Recursive sequences must have an initial term or terms as starting points to work.
For the sequence provided, \(a_{n+1} = \frac{1}{2}a_n + 2\), the formula tells us what the next sequence term will be using the current term \(a_n\). Starting with an initial term, like \(a_0\), you can calculate \(a_1\), then \(a_2\), and so on. Recursive sequences often reveal powerful patterns of growth or decay, depending on how the recursive formula is constructed.
Other exercises in this chapter
Problem 99
The sequence \(\mid a_{n}\) \\} is recursively defined. Compute \(a_{n}\) for \(n=1,2, \ldots, 5\) $$ a_{n+1}=a_{n}+\frac{1}{a_{n}}, a_{0}=1 $$
View solution Problem 100
The sequence \(\mid a_{n}\) \\} is recursively defined. Compute \(a_{n}\) for \(n=1,2, \ldots, 5\) $$ a_{n+1}=2 a_{n}^{2}, a_{0}=1 $$
View solution Problem 102
The sequence \(\left\\{a_{n} \mid\right.\) 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 103
The sequence \(\left\\{a_{n} \mid\right.\) is recursively defined. Find all fixed points of \(\left[a_{n}\right\\} .\) $$ a_{n+1}=\frac{5}{2}-\frac{1}{2} a_{n}
View solution