11.0 Intro to Sequences

Overview of Sequences

Sequences are ordered lists of numbers that follow a specific pattern or rule. In calculus and higher mathematics, understanding sequences is fundamental for studying series, limits, and convergence. This section introduces the basic concepts, formulas, and examples related to sequences.

• Objective 1: List the first several terms of a sequence

• Objective 2: Find the (nth term) formula for a sequence

• Objective 3: Find the recursive formula for a sequence

• Objective 4: Evaluate factorial expressions

Terms of a Sequence

The terms of a sequence are the individual elements, usually denoted as , where is the position in the sequence. Sequences can be defined by explicit (nth term) formulas or recursive formulas.

(Nth Term) Formula

• Definition: An explicit formula gives the value of the nth term directly as a function of .

Recursive Formula

• Definition: A recursive formula defines each term based on previous terms.

• Example 1: , First five terms:

• Example 2: , First five terms:

Fibonacci Sequence

The Fibonacci sequence is a famous recursive sequence where each term is the sum of the two preceding terms.

• Formula: , ,

• First five terms:

Finding the nth Term Formula

Some sequences are given as a list of terms, and the task is to find a general formula for the nth term.

• Example:

• Pattern: Numerators increase by 1, denominators are powers of 5, and the sign alternates.

• General formula:

• Additional info: The sign alternates due to , and the numerator follows based on the sequence's pattern.

Evaluating Factorial Expressions

Factorials are often used in sequences and series. The factorial of a positive integer is the product of all positive integers less than or equal to .

• Definition:

• Example:

Summary Table: Explicit vs. Recursive Formulas

Key Points

• Sequences can be defined explicitly or recursively.

• Finding the nth term helps in analyzing and predicting sequence behavior.

• Factorials and special sequences (like Fibonacci) are important in calculus and discrete mathematics.