Lecture Notes: Arithmetic and Geometric Sequences

Introduction

• Focus on arithmetic sequences.
• Distinction between arithmetic and geometric sequences.

Definitions and Examples

Arithmetic Sequence

• Example: 3, 7, 11, 15, 19, 23, 27
• Common Difference: The constant difference between consecutive terms (e.g., +4).

Geometric Sequence

• Example: 3, 6, 12, 24, 48, 96, 192
• Common Ratio: The constant factor between consecutive terms (e.g., ×2).

Arithmetic vs Geometric Sequences

• Arithmetic: Pattern based on addition/subtraction.
• Geometric: Pattern based on multiplication/division.

Calculating Means

Arithmetic Mean

• Formula: (a + b) / 2
• Example: For 3 and 11 → (3 + 11) / 2 = 7

Geometric Mean

• Formula: √(a × b)
• Example: For 3 and 12 → √(3 × 12) = √36 = 6

Formulas for nth Term

Arithmetic Sequence

• Formula: aₙ = a₁ + (n - 1)d
• Example: To find the 5th term: a₅ = 3 + (5 - 1) × 4 = 19

Geometric Sequence

• Formula: aₙ = a₁ × rⁿ⁻¹
• Example: To find the 6th term: a₆ = 3 × 2⁵ = 96

Partial Sums

Arithmetic Sequence

• Formula: Sₙ = (a₁ + aₙ) / 2 × n
• Example: Sum of first 7 terms: S₇ = (3 + 27) / 2 × 7 = 105

Geometric Sequence

• Formula: Sₙ = a₁ (1 - rⁿ) / (1 - r)
• Example: Sum of first 6 terms: S₆ = 3 × (1 - 2⁶) / (1 - 2) = 189

Sequence vs Series

• Sequence: List of numbers.
• Series: Sum of the numbers in a sequence.
• Infinite vs finite sequences/series.

Identifying Patterns

1. Determine if it's a sequence (list) or series (sum).
2. Identify if it's finite (ends) or infinite (continues).
3. Check for arithmetic (common difference) or geometric (common ratio).

Practice Problems

1. Describe pattern and identify type (sequence/series, finite/infinite, arithmetic/geometric).
2. Calculate first few terms using given formulas.
3. Use nth term formulas to generate terms and find sums.

Examples

• Example Sequences & Series: Arithmetic (common difference) vs Geometric (common ratio).
• Recursive formulas: Use previous term to find the next.

Sum Formulas

• Use partial sum formulas to calculate the sum of a specific number of terms for both arithmetic and geometric sequences.

Additional Practice

• Determine if a given sequence is arithmetic or geometric.
• Write explicit formulas for given sequences.
• Calculate specific terms and sums for both types of sequences.