Sequences and Series

Arts 1–4

Foreign Languages 1–4

Explores arithmetic and geometric sequences and series including sigma notation and sums to infinity.

📘 Topic Summary

Sequences and series are fundamental concepts in mathematics that help us understand patterns and relationships between numbers. This study guide will explore arithmetic and geometric sequences, as well as series, including the use of sigma notation and sums to infinity.

📖 Glossary

Arithmetic Sequence: A sequence where each term is obtained by adding a fixed constant to the previous term.
Geometric Sequence: A sequence where each term is obtained by multiplying the previous term by a fixed constant.
Sigma Notation: A mathematical notation used to represent the sum of a sequence or series.
Sum to Infinity: The concept of adding up an infinite number of terms in a sequence or series.

⭐ Key Points

Sequences and series are used to model real-world phenomena, such as population growth and financial transactions.
Arithmetic sequences can be used to find the sum of a finite number of terms using the formula (n/2)(a1 + an).
Geometric sequences can be used to find the sum of a finite number of terms using the formula (a1 * (r^(n-1))) / (r - 1), where r is the common ratio.
Sigma notation is often used to represent the sum of a sequence or series, making it easier to work with infinite sums.
Sequences and series can be used to solve problems in fields such as physics, engineering, and economics.

🔍 Subtopics

Introduction to Sequences

A sequence is a set of numbers in a specific order, where each term is determined by a rule or formula. For example, the sequence 2, 5, 8, 11, ... is an arithmetic sequence because each term increases by 3. A sequence can be finite or infinite.

Arithmetic Sequences

An arithmetic sequence is a sequence where each term is obtained by adding a fixed constant to the previous term. The common difference, d, is the constant added to get from one term to the next. For example, the sequence 2, 5, 8, 11, ... has a common difference of 3.

Geometric Sequences

A geometric sequence is a sequence where each term is obtained by multiplying the previous term by a fixed constant. The common ratio, r, is the constant multiplied to get from one term to the next. For example, the sequence 2, 6, 18, 54, ... has a common ratio of 3.

Sigma Notation

Sigma notation is a shorthand way to represent the sum of a finite number of terms in a sequence. It is denoted by Σ and is read as 'summation'. For example, Σn=1 to 5 n^2 represents the sum of the squares of the first five positive integers.

Sums to Infinity

A sum to infinity is a sum that extends indefinitely. It can be represented using sigma notation as Σn=1 to ∞. For example, Σn=1 to ∞ (1/n) represents the sum of the reciprocals of all positive integers.

Real-World Applications

Sequences and series have many real-world applications. For instance, population growth can be modeled using geometric sequences, while interest rates on savings accounts can be calculated using arithmetic sequences.

Common Mistakes to Avoid

When working with sequences and series, it is easy to make mistakes such as miscounting the number of terms or incorrectly applying sigma notation. To avoid these errors, it is essential to carefully read and understand the problem before attempting to solve it.

Practice Problems

1. Find the next three terms in the arithmetic sequence: 2, 5, 8, 11, ... 2. Determine whether the geometric sequence: 2, 6, 18, 54, ... is increasing or decreasing.

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