Sequences of Numbers

A sequence is a list of numbers in a specific order. Each number in the sequence is called a term. The terms in a sequence are usually generated according to a pattern or rule. Sequences can be either finite (with a specific number of terms) or infinite (continuing indefinitely).

Types of Sequences

There are different types of sequences, including:

1. Arithmetic Sequence: In an arithmetic sequence, each term is obtained by adding a constant value to the previous term. The constant value is called the common difference. The general form of an arithmetic sequence is: a_n = a₁ + (n-1)d, where a_n is the n-th term, a₁ is the first term, and d is the common difference.
2. Geometric Sequence: In a geometric sequence, each term is obtained by multiplying the previous term by a constant value. The constant value is called the common ratio. The general form of a geometric sequence is: a_n = a₁ * r^(n-1), where a_n is the n-th term, a₁ is the first term, and r is the common ratio.
3. Fibonacci Sequence: The Fibonacci sequence is a special sequence where each term is the sum of the two preceding terms, starting with 0 and 1. The sequence begins: 0, 1, 1, 2, 3, 5, 8, 13, 21, and so on.

Common Notation

When working with sequences, it's important to understand common notation used to represent them:

• a_n: The n-th term of the sequence.
• a₁: The first term of the sequence.
• d: The common difference in an arithmetic sequence.
• r: The common ratio in a geometric sequence.

Study Guide

To study sequences of numbers, it's important to:

1. Understand the differences between arithmetic, geometric, and other types of sequences.
2. Be able to identify the pattern or rule used to generate the terms of a sequence.
3. Know how to find specific terms in a sequence, given the sequence type and its initial terms or common ratio/difference.
4. Practice recognizing and generating sequences in various contexts, such as real-world problems or mathematical patterns.
5. Explore the properties and behaviors of different types of sequences, including their sums and limits in the case of infinite sequences.

By mastering the concepts and skills related to sequences of numbers, you'll be well-prepared to tackle problems involving patterns, series, and mathematical modeling in various fields.