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.


a.Find the first five terms a₀, a₁, ..., a₄ of each sequence.


25.aₙ₊₁ = (1 / 2) aₙ + 8;a₀ = 80

Verified step by step guidance

1. Identify the given recurrence relation and initial term: \(a_{n+1} = \frac{1}{2} a_n + 8\) with \(a_0 = 80\).
2. Calculate the first term: \(a_0\) is given as 80.
3. Find the second term by substituting \(n=0\) into the recurrence: \(a_1 = \frac{1}{2} a_0 + 8\).
4. Find the third term by substituting \(n=1\): \(a_2 = \frac{1}{2} a_1 + 8\).
5. Continue this process to find \(a_3 = \frac{1}{2} a_2 + 8\) and \(a_4 = \frac{1}{2} a_3 + 8\).

Key Concepts

Recurrence Relations

A recurrence relation defines each term of a sequence based on one or more previous terms. In this problem, the sequence is defined recursively by aₙ₊₁ = (1/2) aₙ + 8, meaning each term depends on the previous term. Understanding how to apply the relation iteratively is essential to find specific terms.

Monotonic and Bounded Sequences

A sequence is monotonic if it is either entirely non-increasing or non-decreasing, and bounded if its terms stay within fixed limits. These properties ensure the sequence behaves predictably and often converges, which helps in analyzing long-term behavior and verifying the correctness of computed terms.

Iterative Computation of Sequence Terms

To find specific terms like a₀ through a₄, start with the initial term and repeatedly apply the recurrence relation. This step-by-step calculation is fundamental for understanding the sequence's progression and verifying patterns or limits suggested by the recurrence.