Arithmetic sequence

Definition

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This fixed difference, known as the common difference, allows for a straightforward method to find any term in the sequence. Arithmetic sequences are foundational in understanding linear patterns and can also be contrasted with geometric sequences, where the ratio between terms is constant instead of the difference.

5 Must Know Facts For Your Next Test

1. The general formula for finding the n-th term of an arithmetic sequence is given by $$a_n = a_1 + (n-1) \cdot d$$, where $$a_1$$ is the first term and $$d$$ is the common difference.
2. Arithmetic sequences are used in various real-world applications, including finance for calculating simple interest and in determining evenly spaced schedules.
3. The sum of the first n terms of an arithmetic sequence can be calculated using the formula $$S_n = \frac{n}{2} \cdot (a_1 + a_n)$$ or $$S_n = \frac{n}{2} \cdot (2a_1 + (n-1)d)$$.
4. If the common difference is positive, the arithmetic sequence increases; if negative, it decreases. A zero common difference results in a constant sequence.
5. An arithmetic sequence can be represented graphically as a straight line when plotted, indicating its linear nature.

Review Questions

• How can you determine if a given sequence is an arithmetic sequence? Provide examples of how to identify the common difference.
  • To determine if a given sequence is an arithmetic sequence, check if the difference between consecutive terms is constant. For instance, in the sequence 2, 5, 8, 11, the differences are all 3 (5-2, 8-5, 11-8), confirming it's an arithmetic sequence with a common difference of 3. If any differences vary, then it is not an arithmetic sequence.
• How does the concept of arithmetic sequences relate to geometric sequences, particularly in terms of their definitions and applications?
  • Arithmetic sequences involve adding a constant value to get from one term to another, while geometric sequences multiply by a constant value. This fundamental difference shapes their applications: arithmetic sequences often model situations involving uniform growth or decrease, like salary increments, while geometric sequences might represent scenarios like population growth or interest compounding. Understanding both types helps in analyzing different mathematical models and real-world phenomena.
• Evaluate how knowledge of arithmetic sequences can aid in solving problems involving financial calculations and budgeting.
  • Understanding arithmetic sequences is crucial for financial calculations such as determining total savings over time with regular deposits or calculating interest earned at a fixed rate over time. For example, if you save $100 each month (an arithmetic sequence), you can easily calculate your total savings after n months using the sum formula. This knowledge equips individuals to budget effectively and plan for future financial goals by recognizing predictable patterns in their savings or expenses.

Related terms

common difference: The constant amount added to each term in an arithmetic sequence to obtain the next term.

n-th term: The term located at position n in a sequence, which can be calculated using a specific formula for arithmetic sequences.

geometric sequence: A sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.