49–50. Limits from graphs Consider the following sequences. Find the first four terms of the sequence .Based on part (a) and the figure, determine a plausible limit of the sequence.
aₙ = 2 + 2⁻ⁿ;n = 1, 2, 3, …

Verified step by step guidance

1. Identify the sequence given: \(a_n = 2 + 2^{-n}\) for \(n = 1, 2, 3, \ldots\).
2. Calculate the first four terms by substituting \(n = 1, 2, 3, 4\) into the formula: - \(a_1 = 2 + 2^{-1}\) - \(a_2 = 2 + 2^{-2}\) - \(a_3 = 2 + 2^{-3}\) - \(a_4 = 2 + 2^{-4}\)
3. Evaluate each term by calculating the powers of 2 with negative exponents (e.g., \(2^{-1} = \frac{1}{2}\), \(2^{-2} = \frac{1}{4}\), etc.) and then add 2 to each result to find the numerical values of the first four terms.
4. Observe the graph which shows the sequence values plotted against \(n\). Notice that as \(n\) increases, the points approach a horizontal line near \(a_n = 2\).
5. Based on the formula and the graph, conclude that the plausible limit of the sequence as \(n\) approaches infinity is \(2\), because \$2^{-n}\( approaches 0, making \)a_n\( approach \)2 + 0 = 2$.

Key Concepts

Sequences and Terms

A sequence is an ordered list of numbers defined by a specific formula for its terms. Each term is identified by its position n, and understanding how to compute the first few terms helps reveal the sequence's behavior. For example, aₙ = 2 + 2⁻ⁿ generates terms approaching a limit as n increases.

Limits of Sequences

The limit of a sequence is the value that the terms approach as n becomes very large. If the terms get closer and closer to a fixed number, that number is the sequence's limit. This concept helps describe long-term behavior, such as aₙ approaching 2 in the given sequence.

Interpreting Graphs of Sequences

Graphs visually represent sequence terms as points, showing trends and convergence. By examining the plotted points, one can estimate the limit and verify calculations. In the provided graph, the points approach the horizontal line at aₙ = 2, indicating the sequence's limit.