Geometric Sequences and Series

Introduction

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

Geometric Sequence

• Example: 3, 6, 12, 24, 48...
• Definition: A geometric sequence has a common ratio.
• Common Ratio:
  • Found by dividing consecutive terms.
  • Example: 6/3 = 2, 12/6 = 2.
• Comparison to Arithmetic Sequence:
  • Arithmetic sequence has a common difference.
  • Example: 8 - 5 = 3, 11 - 8 = 3.
• Operations:
  • Arithmetic: Addition/Subtraction.
  • Geometric: Multiplication/Division.

Geometric Series

• Definition: The sum of a geometric sequence.
• Example: 3 + 6 + 12 + 24 + 48...
• Formula for the n-th term (a_n):
  • a_n = a_1 * r^(n - 1)
  • Example: Finding the 5th term: 3 * 2^(5-1) = 48.

Partial Sum Formula

• Formula for the partial sum of a geometric series (S_n):
  • S_n = a_1 * (1 - r^n) / (1 - r)
• Example: Sum of the first 5 terms:
  • S_5 = 3 * (1 - 2^5) / (1 - 2) = 93.

Infinite Geometric Series

• Definition: A series that continues indefinitely.
• Convergence:
  • Applies when |r| < 1.
• Sum Formula for infinite series:
  • S_infinity = a_1 / (1 - r)
• Example:
  • For series with terms 8, 4, 2, 1, 1/2...
  • r = 1/2 (since |r| < 1), sum = 8 / (1 - 1/2) = 16.

Arithmetic Mean vs. Geometric Mean

• Arithmetic Mean (M_A): Average of two numbers.
• Geometric Mean (M_G): Square root of the product of two numbers.
• Example:
  • Arithmetic mean between 5 and 11: (5 + 11)/2 = 8.
  • Geometric mean between 3 and 12: sqrt(3 * 12) = 6.*

Formulas for Geometric Sequences

• To relate terms within a geometric sequence:
  • For a_n = a_m * r^(n-m)
  • Example: To relate the 5th and 2nd terms, multiply by r^3.*

Identifying Sequences and Series

• Determine if they are arithmetic or geometric, finite or infinite.
• Example: Common difference indicates arithmetic; common ratio indicates geometric.

Practice Problems

• Writing terms of geometric sequences:
  • Example 1: First term 2, common ratio 3; terms are 2, 6, 18, 54, 162.
  • Example 2: First term 80, common ratio 1/2; terms are 80, 40, 20, 10, 5.

Summary of Key Concepts

• Finite Series: Use formula for partial sums.
• Infinite Series: Check convergence, use infinite sum formula.
• Finding nth Term: Use geometric sequence formula.

Conclusion

• Understand definitions and formulas for geometric sequences and series.
• Apply knowledge to solve problems involving both finite and infinite series.