Sequences of Numbers

A sequence is a list of numbers in a specific order. Each number in the sequence is called a term. Sequences can be finite or infinite.

Types of Sequences

1. Arithmetic Sequence: In an arithmetic sequence, each term after the first is found by adding the same number to the previous term. This number 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 after the first is found by multiplying the previous term by the same number. This number 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 series of numbers in which each number is the sum of the two preceding ones, usually starting with 0 and 1. The sequence begins 0, 1, 1, 2, 3, 5, 8, 13, and so on.

Study Guide

When studying sequences of numbers, it's important to understand the difference between arithmetic and geometric sequences, as well as how to find specific terms within a sequence.

Key Concepts:

1. Identifying whether a given sequence is arithmetic or geometric.
2. Finding the common difference or common ratio for a given sequence.
3. Finding specific terms within a sequence, given the first term and the common difference or common ratio.
4. Understanding the properties and applications of the Fibonacci sequence.

Practice Problems:

1. Determine whether the following sequences are arithmetic, geometric, or neither:

• a. 3, 6, 9, 12, 15, ...
• b. 2, 4, 8, 16, 32, ...
• c. 1, 1, 2, 3, 5, 8, 13, ...

2. Find the common difference or common ratio for the following sequences:

• a. 7, 10, 13, 16, 19, ...
• b. 5, 10, 20, 40, 80, ...

3. Determine the 10th term of each of the sequences in question 2.

4. Explore the properties of the Fibonacci sequence and identify any patterns or relationships within the sequence.

By understanding and practicing these key concepts and problems, you will build a strong foundation in working with sequences of numbers.