key term - Arithmetic Series

Definition

An arithmetic series is a sequence of numbers where the difference between each consecutive term is constant. It is a type of series that follows a linear pattern and can be used to model various real-world phenomena, such as the growth of a savings account or the depreciation of an asset.

5 Must Know Facts For Your Next Test

1. The formula for the nth term of an arithmetic series is $a_n = a_1 + (n-1)d$, where $a_1$ is the first term and $d$ is the common difference.
2. The formula for the sum of the first $n$ terms of an arithmetic series is $S_n = \frac{n}{2}[2a_1 + (n-1)d]$.
3. Arithmetic series are often used to model linear growth or decay, such as the growth of a savings account or the depreciation of an asset.
4. The common difference in an arithmetic series can be positive, negative, or zero, corresponding to an increasing, decreasing, or constant sequence, respectively.
5. Arithmetic series are a fundamental concept in calculus and are used to approximate the area under a curve or the volume of a three-dimensional object.

Review Questions

• Explain the relationship between an arithmetic series and an arithmetic progression.
  • An arithmetic series and an arithmetic progression are closely related concepts. An arithmetic progression is a sequence of numbers where the difference between each consecutive term is constant, while an arithmetic series is the sum of the terms in an arithmetic progression. In other words, an arithmetic series is the cumulative sum of an arithmetic progression. Both share the common property of a constant difference between terms, which is a defining characteristic of linear patterns.
• Describe how the formulas for the nth term and the sum of the first n terms of an arithmetic series are derived.
  • The formula for the nth term of an arithmetic series, $a_n = a_1 + (n-1)d$, is derived by recognizing that each successive term in the series is obtained by adding the common difference $d$ to the previous term. The formula for the sum of the first $n$ terms, $S_n = \frac{n}{2}[2a_1 + (n-1)d]$, is derived by recognizing that the sum of an arithmetic series can be expressed as the average of the first and last terms multiplied by the number of terms.
• Explain how arithmetic series can be used to model real-world phenomena, and provide examples of such applications.
  • Arithmetic series are often used to model linear growth or decay in real-world scenarios. For example, the growth of a savings account over time can be modeled as an arithmetic series, where the constant difference represents the regular deposits or withdrawals. Similarly, the depreciation of an asset, such as a vehicle, can be modeled as a decreasing arithmetic series, where the common difference represents the constant rate of depreciation. Other applications of arithmetic series include modeling the repayment of a loan, the accumulation of interest, and the growth of a population over time.

Related terms

Sequence:

A sequence is an ordered list of numbers or other mathematical objects that follow a specific pattern.

Common Difference:

The common difference is the constant value that is added or subtracted to obtain each successive term in an arithmetic series.

Arithmetic Progression: An arithmetic progression is a sequence of numbers where the difference between each consecutive term is constant, similar to an arithmetic series.