25–26. Recursively defined sequences
The following sequences {aₙ} from n = 0 to ∞ are defined by a recurrence relation. Assume each sequence is monotonic and bounded.

b.Determine the limit of each sequence.

25.aₙ₊₁ = (1 / 2) aₙ + 8;a₀ = 80

검증된 단계별 안내

Identify the recurrence relation given: \(a_{n+1} = \frac{1}{2} a_n + 8\) with initial value \(a_0 = 80\).

Assuming the sequence converges to a limit \(L\), use the property that as \(n \to \infty\), \(a_n\) and \(a_{n+1}\) both approach \(L\).

Set up the equation for the limit by substituting \(a_n = L\) and \(a_{n+1} = L\) into the recurrence relation: \(L = \frac{1}{2} L + 8\).

Solve the resulting algebraic equation for \(L\) to find the limit of the sequence.

Verify that the sequence is monotonic and bounded to confirm that the limit found is valid.

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주요 개념

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Recursively Defined Sequences

A recursively defined sequence is one where each term is defined based on one or more previous terms using a recurrence relation. Understanding how to express terms in terms of earlier ones is essential for analyzing the sequence's behavior and finding limits.

Monotonicity and Boundedness

A sequence is monotonic if it is either non-increasing or non-decreasing, and bounded if its terms lie within fixed upper and lower limits. These properties guarantee the existence of a limit for the sequence, which is crucial when determining convergence.

Finding Limits of Linear Recurrence Relations

For linear recurrence relations like aₙ₊₁ = r aₙ + c, the limit can be found by setting the limit L equal to its own recurrence: L = rL + c. Solving this equation gives the sequence's limit, assuming |r| < 1 to ensure convergence.

25–26. Recursively defined sequencesThe following sequences {aₙ} from n = 0 to ∞ are defined by a recurrence relation. Assume each sequence is monotonic and bounded.b.Determine the limit of each sequence.25.aₙ₊₁ = (1 / 2) aₙ + 8;a₀ = 80