Lecture on Sequences: Arithmetic and Geometric

Introduction

• Discussion on sequences: Arithmetic and Geometric
• Importance and future applications

Arithmetic Sequence

• Definition: Sequence of numbers where each term is obtained by adding a constant amount to the previous term
• Example: 3, 5, 7, 9 (odd numbers, adding 2 each time)
• Terminology:
  • Constant Difference (D): Difference between terms (e.g., D = 2)
  • First Term (A): Starting term of the sequence
• Formula for nth term: a_n = A + D(n-1)
• Example Calculation:
  • First term A = 2, Difference D = 3
  • Finding the nth term: a_n = 2 + 3(n - 1)
  • Finding the 5th term: Same process yields 14
• System of Equations Example:
  • Given: 4th term = 30, 11th term = 107
  • Create equations: A + 3D = 30, A + 10D = 107
  • Solve for A and D
  • First term A = -3, Difference D = 11
  • First five terms: -3, 8, 19, 30, 41
• Sum of the first n terms (S_n): S_n = n/2 * (2A + (n-1)D)
  • Example: Sum of the first 20 terms of the sequence 3, 10, 17, ...
  • Calculation: 1390
• Application Example:
  • Salary Sequence with $30,000 starting salary, $2300 yearly raise
  • Calculate total earnings over the first 10 years: $403,500*

Geometric Sequence

• Definition: Sequence of numbers where each term is obtained by multiplying the previous term with a constant factor
• Example: First term A, ratio R
• Terminology:
  • Common Ratio (R): Ratio between terms
  • First Term (A): Starting term of the sequence
• Formula for nth term: a_n = A * R^(n-1)
• Example Calculation:
  • First term A = -6, Ratio R = 3
  • Finding first four terms: -6, -18, -54, -162
  • nth term formula: a_n = -6 * 3^(n-1)
• Application Example:
  • Bouncing ball with initial height of 80 feet, rebounds to 3/4 of previous height
  • Model with geometric sequence: 80, 60, 45, 33.75, 25.31, 18.98
  • Calculation for height after the 5th bounce
• Identifying Geometric Sequences:
  • Various examples to determine if sequences are geometric
  • Calculation of common ratios
• Partial Sum of Geometric Sequence:
  • Formula: S_n = A * (1 - R^n) / (1 - R) for R ≠ 1
  • Example: First term A = 3, 3rd term = 4/3,
  • Find 5th term and sum of first 5 terms*

Practical Problems and Formulas

• Determining Sequence Types: Arithmetic vs. Geometric
  • Various examples given
  • Identify constants, ratios, and general patterns
• Arithmetic Sequence Examples
• Geometric Sequence Examples
• Partial Sums: Derivation and applications