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.


aₙ₊₁ = 2aₙ(1 − aₙ);a₀ = 0.3

Verified step by step guidance

1. Step 1: Understand the recurrence relation given: \(a_{n+1} = 2a_n(1 - a_n)\) with initial term \(a_0 = 0.3\). This defines each term based on the previous term.
2. Step 2: Calculate the first three terms explicitly to observe the behavior of the sequence: compute \(a_1 = 2a_0(1 - a_0)\), then \(a_2 = 2a_1(1 - a_1)\), and \(a_3 = 2a_2(1 - a_2)\). 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: To find the limit analytically, assume the sequence converges to a limit \(L\). Since the sequence converges, the limit satisfies the same recurrence relation in the steady state: \(L = 2L(1 - L)\).
4. Step 4: Solve the equation \(L = 2L(1 - L)\) for \(L\). This will involve rearranging the equation into a quadratic form and finding the roots.
5. Step 5: Analyze the roots found in Step 4 and use the initial condition and monotonicity from Step 2 to determine which root is the actual limit of the sequence.

Key Concepts

Recurrence Relations

A recurrence relation defines each term of a sequence using previous terms. Understanding how to generate terms from the initial value is essential to analyze the sequence's behavior, such as monotonicity and convergence.

Monotonicity and Boundedness

A sequence is monotonic if it is either nonincreasing or nondecreasing throughout. If it is also bounded (confined within limits), these properties guarantee convergence, meaning the sequence approaches a finite limit.

Finding Limits of Sequences Analytically

To find the limit of a sequence defined by a recurrence, set the limit L equal to the expression defining the next term in terms of L. Solving this fixed-point equation helps determine the sequence's long-term behavior.