Arithmetic sequence

Definition

An arithmetic sequence is a list of numbers in which the difference between consecutive terms is constant. This common difference can be positive, negative, or zero, and it plays a crucial role in determining the behavior of the sequence as it progresses. Understanding arithmetic sequences helps to analyze their monotonic properties, limits, and foundational aspects of sequences more broadly.

5 Must Know Facts For Your Next Test

1. In an arithmetic sequence, if the first term is 'a' and the common difference is 'd', the n-th term can be expressed as 'a + (n - 1)d'.
2. Arithmetic sequences can be both monotonic increasing and monotonic decreasing depending on whether the common difference is positive or negative.
3. The sum of the first n terms of an arithmetic sequence can be calculated using the formula: $$S_n = \frac{n}{2} (2a + (n - 1)d)$$.
4. The limit of an arithmetic sequence does not exist if the common difference is non-zero, as it diverges to infinity or negative infinity.
5. A special case occurs when the common difference is zero; in this case, all terms are equal, resulting in a constant sequence.

Review Questions

• How do you determine whether an arithmetic sequence is monotonic?
  • To determine if an arithmetic sequence is monotonic, you need to look at the common difference. If the common difference is positive, the sequence is monotonically increasing because each term is larger than the previous one. Conversely, if the common difference is negative, the sequence is monotonically decreasing since each term is smaller than its predecessor. If the common difference is zero, the sequence remains constant.
• Explain how to find the limit of an arithmetic sequence and why it behaves differently based on its common difference.
  • An arithmetic sequence will diverge unless its common difference is zero. If the common difference is non-zero, as n approaches infinity, the terms will either increase without bound or decrease without bound towards negative infinity, meaning there is no limit. However, if the common difference is zero, every term is identical, and thus the limit exists and equals that constant value.
• Evaluate the significance of arithmetic sequences in understanding broader mathematical concepts like convergence and series.
  • Arithmetic sequences serve as a fundamental building block for understanding more complex mathematical concepts such as convergence and series. While they themselves do not converge if their common difference is non-zero, they lay the groundwork for exploring how sequences behave under different conditions. The study of their sums leads to series, providing insights into how infinite sequences can converge to finite values when altered, such as in geometric sequences. This connection highlights important relationships in mathematics that influence various fields like calculus and numerical analysis.

Related terms

Common Difference: The constant amount that each term in an arithmetic sequence increases or decreases by, denoted as 'd'.

Geometric Sequence: A sequence where each term is found by multiplying the previous term by a fixed, non-zero number called the common ratio.

Convergence:

The property of a sequence to approach a specific value or limit as the number of terms increases indefinitely.