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

Verified step by step guidance

1. Identify the recurrence relation given: \(a_{n+1} = \frac{1}{2} a_n + 8\) with initial value \(a_0 = 80\).
2. Assuming the sequence converges to a limit \(L\), use the property that as \(n \to \infty\), \(a_n\) and \(a_{n+1}\) both approach \(L\).
3. Set up the equation for the limit by substituting \(a_n = L\) and \(a_{n+1} = L\) into the recurrence relation: \(L = \frac{1}{2} L + 8\).
4. Solve the resulting algebraic equation for \(L\) to find the limit of the sequence.
5. Verify that the sequence is monotonic and bounded to confirm that the limit found is valid.

Key Concepts

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.