Math Lesson 5: Number Patterns, Sequences, and Series

5.1 Number Patterns

Number patterns are sequences of numbers that follow a specific rule or relationship. This rule might involve adding, subtracting, multiplying, dividing, or a combination of these operations.

Example:

The sequence 2, 4, 6, 8, 10... follows the rule "add 2 to the previous number".

5.2 Arithmetic Sequences

An arithmetic sequence is a number pattern where the difference between consecutive terms is constant. This constant difference is called the common difference (d).

Formula: a_n = a₁ + (n - 1)d

where:

• a_n is the nth term of the sequence
• a₁ is the first term
• n is the term number
• d is the common difference

5.3 Quadratic Sequences

A quadratic sequence is a sequence where the second differences between consecutive terms are constant. This means that the difference between consecutive terms is changing at a constant rate.

Formula: a_n = an² + bn + c

where:

• a_n is the nth term of the sequence
• a, b, and c are constants
• n is the term number

Example:

The sequence 1, 4, 9, 16, 25... is a quadratic sequence with a common second difference of 2.

5.4 Geometric Sequences

A geometric sequence is a sequence where each term is found by multiplying the previous term by a constant factor. This factor is called the common ratio (r).

Formula: a_n = a₁ * r^{(n - 1)}

where:

• a_n is the nth term of the sequence
• a₁ is the first term
• n is the term number
• r is the common ratio

5.5 Arithmetic and Geometric Series

A series is the sum of the terms in a sequence. We can find the sum of an arithmetic or geometric series using specific formulas.

Arithmetic Series

Formula: S_n = (n/2) * [2a₁ + (n-1)d]

where:

• S_n is the sum of the first n terms
• a₁ is the first term
• n is the number of terms
• d is the common difference

Geometric Series

Formula: S_n = a₁ * (1 - rⁿ) / (1 - r)

where:

• S_n is the sum of the first n terms
• a₁ is the first term
• n is the number of terms
• r is the common ratio

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