Problem 93

Question

In Exercises $91-98$ , assume that each sequence converges and find its limit. $$ a_{1}=-4, \quad a_{n+1}=\sqrt{8+2 a_{n}} $$

Step-by-Step Solution

Verified

Answer

The limit of the sequence is 4.

1Step 1: Understand the Sequence Definition

The sequence is defined recursively by $ a_1 = -4 $ and $ a_{n+1} = \sqrt{8 + 2a_n} $. Each term depends on the previous one, and we are asked to find the limit as $ n \to \infty $.

2Step 2: Assume the Limit Exists

Assume that the sequence converges to a limit $ L $. That means for large $ n $, both $ a_n $ and $ a_{n+1} $ approach $ L $.

3Step 3: Set Up the Limit Equation

Since the limit of both $ a_n $ and $ a_{n+1} $ is $ L $, substitute $ L $ into the recursive formula: $ L = \sqrt{8 + 2L} $.

4Step 4: Solve the Limit Equation

Square both sides to eliminate the square root: $ L^2 = 8 + 2L $. Rearrange this equation to standard quadratic form: $ L^2 - 2L - 8 = 0 $.

5Step 5: Solve the Quadratic Equation

Use the quadratic formula $ L = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $ where $ a = 1 $, $ b = -2 $, $ c = -8 $. Plug in the values: $ L = \frac{2 \pm \sqrt{4 + 32}}{2} = \frac{2 \pm \sqrt{36}}{2} = \frac{2 \pm 6}{2} $.

6Step 6: Choose the Appropriate Root

The solutions to the quadratic equation are $ L = 4 $ and $ L = -2 $. Since the sequence terms involve a square root, which only produces non-negative results, $ L = -2 $ cannot be correct if the sequence ever enters non-negative real numbers (which occurs since starting at $ -4 $ and iteratively applying the formula heads towards positive domain). Thus, $ L = 4 $.

7Step 7: Verify Convergence with Initial Terms

Substitute initial values. Starting from $ a_1 = -4 $, the next terms move towards positive values, as each iteration using $ a_n = \sqrt{8 + 2a_{n-1}} $ computes to an increasingly positive $ a_{n+1} $. This supports convergence to $ L = 4 $.

Key Concepts

Recursive sequencesConvergence of sequencesQuadratic equations

Recursive sequences

A recursive sequence is one where each term is defined in terms of previous terms. It relies on a starting point, often called the initial condition, and a repeated process to generate new terms.
For example, if we know $ a_1 = -4 $ and $ a_{n+1} = \sqrt{8 + 2a_n} $, we can calculate each subsequent term using the value of the previous one. This type of calculation is like walking step by step towards the sequence's behavior, one step at a time.
Recursive sequences are very useful because they can model processes or phenomena that have inherent repetition or dependency in their structure:

Each term is constructed by applying the sequence's rule repeatedly.
Understanding the first few terms gives insight into the behavior of the entire sequence.
It provides a foundation for more advanced concepts like convergence.

Although recursive sequences can initially seem complex, breaking them down into their components by looking at each term separately can make understanding them straightforward.

Convergence of sequences

The convergence of a sequence refers to the behavior of a sequence as its terms progress towards infinity. A sequence is said to converge if it approaches a specific value, called the limit, as more terms are added.
To determine if a sequence like the one given in the exercise converges, we use the property that if convergent, it will have the same value for both $ a_n $ and $ a_{n+1} $ when $ n $ is large. Let's assume it converges to $ L $. Then:

$ a_n \to L $ and $ a_{n+1} \to L $, meaning as $ n \to \infty $, all terms move towards $ L $.
Using this property, substitute $ L $ into the recursive formula to find the limit equation.

If a limit exists, it demonstrates convergence. Our task often highlights finding these suspected limits and then checking if the entire sequence behaves as predicted to validate this limit's correctness. Converging sequences have crucial applications in real-life scenarios, especially in predicting steady states or balanced outcomes.

Quadratic equations

Quadratic equations appear frequently in various mathematical contexts, particularly when solving for limits in recursive sequences. A standard quadratic equation is expressed in the form $ ax^2 + bx + c = 0 $. Solving a quadratic equation often involves finding values for $ x $ that satisfy the equation.
In our case, after assuming the sequence's limit exists as $ L $, we derived a quadratic equation: $ L^2 - 2L - 8 = 0 $. To solve it, we used the quadratic formula:

$ L = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $.
This formula calculates the possible solutions for $ L $.
Understanding which solution fits the context, only non-negative roots were considered since our sequence eventually reached a positive number via square roots.

Quadratic equations not only provide us necessary solutions in sequences and limits but also appear in various real-world applications like physics for trajectory paths, economics for profit functions, and more. Efficiently solving these offers a crucial tool in both mathematical exploration and practical implementation.

Problem 93

Problem 94

Other exercises in this chapter

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Problem 94

In Exercises $91-98$ , assume that each sequence converges and find its limit. $$ a_{1}=0, \quad a_{n+1}=\sqrt{8+2 a_{n}} $$

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