Sequences

What is a Sequence?

• A sequence is an ordered list of numbers, often following a specific pattern or rule.

• Each number in the sequence is called a term.

Types of Sequences

• Arithmetic Sequences:

  • Each term is found by adding a constant value (called the common difference) to the previous term.

  • Example: 2, 5, 8, 11, 14, ... (common difference: 3)

• Geometric Sequences:

  • Each term is found by multiplying the previous term by a constant value (called the common ratio).

  • Example: 3, 6, 12, 24, 48, ... (common ratio: 2)

• Other Types:

  • Fibonacci Sequence: 1, 1, 2, 3, 5, 8, 13, ... (each term is the sum of the two preceding terms)

  • Factorial Sequence: 1, 2, 6, 24, 120, ... (n! = n × (n-1) × (n-2) × ... × 1)

  • Square Number Sequence: 1, 4, 9, 16, 25, ... (n²)

Key Concepts

• Terms: The individual numbers in a sequence.

• Explicit Formula: A formula that directly calculates the nth term of a sequence.

• Recursive Formula: A formula that defines the nth term in terms of previous terms.

• Series: The sum of the terms in a sequence.

Applications of Sequences

• Modeling Growth and Decay: Population growth, compound interest, radioactive decay

• Computer Science: Algorithms, data structures

• Finance: Compound interest, annuities

• Nature: Patterns in nature, such as the Fibonacci sequence in plant growth