51–56. {Use of Tech} Recurrence relations Consider the following recurrence relations. Make a table with at least ten terms and determine a plausible limit of the sequence or state that the sequence diverges.


aₙ₊₁ = 4aₙ + 1 a₀ = 1

Verified step by step guidance

1. Identify the recurrence relation given: \(a_{n+1} = 4a_n + 1\) with initial term \(a_0 = 1\).
2. Create a table of terms by calculating each subsequent term using the recurrence relation: for each \(n\), compute \(a_{n+1}\) by substituting \(a_n\) into the formula \(a_{n+1} = 4a_n + 1\).
3. Calculate the first ten terms step-by-step: start with \(a_0 = 1\), then find \(a_1 = 4 \times a_0 + 1\), \(a_2 = 4 \times a_1 + 1\), and so on until \(a_9\).
4. Observe the behavior of the terms in the table: check if the terms are increasing without bound, approaching a fixed number, or oscillating.
5. Based on the observed pattern, determine whether the sequence converges to a limit or diverges (grows without bound).

Key Concepts

Recurrence Relations

A recurrence relation defines each term of a sequence using previous terms. It provides a way to generate sequences step-by-step, often starting from an initial value. Understanding how to apply and iterate these relations is essential for analyzing the behavior of sequences.

Sequence Limits and Convergence

The limit of a sequence is the value the terms approach as the index goes to infinity. A sequence converges if its terms get arbitrarily close to a finite number; otherwise, it diverges. Determining limits helps understand the long-term behavior of sequences defined by recurrence relations.

Iterative Computation and Pattern Recognition

Computing terms iteratively involves using the recurrence formula repeatedly to build a sequence table. Observing these computed values helps identify patterns or trends, which is crucial for hypothesizing about the sequence’s limit or divergence before formal proof.