45–48. {Use of Tech} Explicit formulas for sequences Consider the formulas for the following sequences {aₙ}ₙ₌₁∞
 Make a table with at least ten terms and determine a plausible limit of the sequence or state that the sequence diverges.
aₙ = ⁿ² + n ;n = 1, 2, 3, …

Verified step by step guidance

1. Identify the given sequence formula: \(a_n = n^2 + n\) for \(n = 1, 2, 3, \ldots\).
2. Create a table of values by substituting \(n\) from 1 to 10 into the formula to find the first ten terms: calculate \(a_1, a_2, \ldots, a_{10}\).
3. Observe the pattern of the terms in the table to see how the sequence behaves as \(n\) increases.
4. Analyze the growth of the terms by considering the dominant term \(n^2\) in the formula, which suggests how the sequence behaves for large \(n\).
5. Based on the behavior of the terms, determine whether the sequence approaches a finite limit or diverges to infinity.

Key Concepts

Sequences and Their Terms

A sequence is an ordered list of numbers defined by a specific formula for its terms. Each term aₙ depends on the index n, which usually starts at 1 and increases by 1. Understanding how to generate terms from the explicit formula is essential for analyzing the sequence's behavior.

Limits of Sequences

The limit of a sequence describes the value that the terms approach as n becomes very large. If the terms get arbitrarily close to a finite number, the sequence converges; otherwise, it diverges. Determining the limit helps understand the long-term behavior of the sequence.

Using Tables and Numerical Approximations

Creating a table of terms allows visualization of the sequence's progression and helps identify patterns or trends. By calculating several terms, one can estimate whether the sequence approaches a finite limit or grows without bound, aiding in conjecturing convergence or divergence.