Arithmetic Sequences and Geometric Sequences

Introduction to Sequences

• Distinction between arithmetic and geometric sequences

Arithmetic Sequence Example

• Sequence: 3, 7, 11, 15, 19, 23, 27
• Common Difference: 4
  • To go from 3 to 7, add 4
  • To go from 7 to 11, add 4

Geometric Sequence Example

• Sequence: 3, 6, 12, 24, 48, 96, 192
• Common Ratio: 2
  • To go from 3 to 6, multiply by 2
  • To go from 6 to 12, multiply by 2

Patterns in Sequences

• Arithmetic Sequence: Based on addition/subtraction
• Geometric Sequence: Based on multiplication/division

Arithmetic Mean vs. Geometric Mean

Arithmetic Mean Calculation

• Formula: (a + b) / 2
• Example 1: Mean of 3 and 11
  • Calculation: (3 + 11) / 2 = 7
• Example 2: Mean of 7 and 23
  • Calculation: (7 + 23) / 2 = 15

Geometric Mean Calculation

• Formula: √(a * b)
• Example 1: Mean of 3 and 12
  • Calculation: √(3 * 12) = 6
• Example 2: Mean of 6 and 96
  • Calculation: √(6 * 96) = 24

Nth Term of Sequences

Arithmetic Sequence Formula

• Formula: a_n = a_1 + (n - 1) * d
  • Example: Find the 5th term for the sequence starting with 3, with a common difference of 4
    • Calculation: a_5 = 3 + (5 - 1) * 4 = 19

Geometric Sequence Formula

• Formula: a_n = a_1 * r^(n - 1)
  • Example: Find the 6th term for the sequence starting with 3, with a common ratio of 2
    • Calculation: a_6 = 3 * 2^(6 - 1) = 96

Partial Sum of Sequences

Arithmetic Series Formula

• Formula: S_n = (a_1 + a_n) / 2 * n
• Example: Sum of first 7 terms for the arithmetic sequence 3, 7, 11, 15, 19, 23, 27
  • Calculation: S_7 = (3 + 27) / 2 * 7 = 105

Geometric Series Formula

• Formula: S_n = a_1 * (1 - r^n) / (1 - r)
• Example: Sum of first 6 terms for the geometric sequence 3, 6, 12, 24, 48, 96
  • Calculation: S_6 = 3 * (1 - 2^6) / (1 - 2) = 189

Sequences vs. Series

• Sequence: A list of numbers
• Series: The sum of numbers in a sequence

Finite vs. Infinite Sequences

• Finite: Has a beginning and an end
• Infinite: Continues indefinitely (indicated by "...")

Practice Problems

Identifying Sequences and Series

• Given numbers, determine if they form a sequence or series, and classify as finite/infinite, arithmetic/geometric/neither.

Finding Next Terms in a Sequence

• Determine common difference for arithmetic sequences to find next terms.

Writing Terms from Recursive Formulas

• Use previous terms to find subsequent terms in recursive definitions.

General Formulas for Sequences

• Use first term and common difference to write explicit formulas for arithmetic sequences.

Conclusion

• Understanding the structure of sequences allows for easy calculation of various terms and sums.
• Methods for identifying and manipulating both arithmetic and geometric sequences are essential in mathematical problem solving.