84–87. {Use of Tech} Sequences by recurrence relations
The following sequences, defined by a recurrence relation, are monotonic and bounded, and therefore converge by Theorem 10.5.


a.Examine the first three terms of the sequence to determine whether the sequence is nondecreasing or nonincreasing.
b.Use analytical methods to find the limit of the sequence.




{Use of Tech}aₙ₊₁ = √(2 + aₙ);a₀ = 3

Verified step by step guidance

1. Step 1: Understand the recurrence relation given: \(a_{n+1} = \sqrt{2 + a_n}\) with initial term \(a_0 = 3\). This defines the sequence where each term depends on the previous term.
2. Step 2: Calculate the first three terms explicitly to observe the behavior of the sequence: compute \(a_1 = \sqrt{2 + a_0}\), then \(a_2 = \sqrt{2 + a_1}\), and \(a_3 = \sqrt{2 + a_2}\). Compare these values to determine if the sequence is nondecreasing (each term greater than or equal to the previous) or nonincreasing (each term less than or equal to the previous).
3. Step 3: Since the sequence is monotonic and bounded, it converges. To find the limit \(L\), assume the sequence converges to \(L\) and use the property that the limit satisfies the recurrence relation: \(L = \sqrt{2 + L}\).
4. Step 4: Solve the equation \(L = \sqrt{2 + L}\) analytically by squaring both sides to eliminate the square root, giving \(L^2 = 2 + L\). Rearrange this into a standard quadratic form: \(L^2 - L - 2 = 0\).
5. Step 5: Solve the quadratic equation \(L^2 - L - 2 = 0\) using the quadratic formula \(L = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) with \(a=1\), \(b=-1\), and \(c=-2\). Then, determine which root is valid by considering the domain and behavior of the sequence.

Key Concepts

Recurrence Relations and Sequences

A recurrence relation defines each term of a sequence based on previous terms. Understanding how to generate terms from the initial value helps analyze the sequence's behavior, such as monotonicity and boundedness, which are crucial for determining convergence.

Monotonicity and Boundedness of Sequences

A sequence is monotonic if it is either nondecreasing or nonincreasing throughout. If a sequence is also bounded (confined within fixed limits), these properties guarantee convergence by the Monotone Convergence Theorem, allowing us to conclude the sequence approaches a finite limit.

Finding Limits of Sequences Using Analytical Methods

To find the limit of a convergent sequence defined by a recurrence, set the limit equal to the expression defining the next term and solve the resulting equation. This method uses the fact that the limit remains unchanged under the recurrence relation, enabling calculation of the exact limit value.