Sequences: Finite and Infinite

Definitions

• Sequence: An ordered list of numbers, considered a function whose domain is the set of positive integers.
• Finite Sequence: A sequence with a last term, represented by 'n' (e.g., 5 terms: 1st, 2nd, 3rd, 4th, 5th).
• Infinite Sequence: A sequence without a last term, typically indicated by three dots (ellipsis).
• Term: Each number in a sequence.

Example Sequences and Patterns

1. Letters Sequence: a, b, g, j
   • Pattern: Skip two letters each time
   • Next terms: m, p, s
2. Odd Numbers: 1, 3, 5, 7
   • Pattern: Odd numbers only
   • Next terms: 9, 11, 13
3. Perfect Squares: 1, 4, 9, 16, 25
   • Pattern: Squares of integers
   • Next terms: 36, 49, 64
4. Arithmetic Sequence: 5, 15, 25, 35
   • Pattern: Add 10 each time
   • Next terms: 45, 55, 65
5. Cumulative Sums: 1, 3, 6, 10, 15, 21, 28
   • Pattern: Add 2, 3, 4, 5, etc., in order
   • Next term calculation: 28 + 8 = 36

General Terms and Formulas

• General Term (Rule): Used to find any term in a sequence
• Examples of using the general term:
  • Sequence: 1, 3, 6, 10, 15
    • General Term: a_n = n(n + 1) / 2
    • Example Calculation for 5th term:
      • a_5 = 5(5 + 1)/2 = 15
  • Sequence: 1, 4, 9, 16, 25
    • General Term: a_n = n^2
    • Example Calculation for 3rd term:
      • a_3 = 3^2 = 9

Finding Patterns and General Terms

• Steps to identify patterns and general terms
  1. Analyze differences/similarities between terms
  2. Identify repetitious operations (e.g., +2, *2, etc.)
  3. Form a general rule based on these observations

Practice Examples

1. Sequence: 1, 8, 27, 64, 125
   • Identified Pattern: Cubes
   • General Term: a_n = n^3
2. Sequence: 1, 1/2, 1/3, 1/4
   • Identified Pattern: Reciprocal of integers
   • General Term: a_n = 1/n
3. Sequence: -5, 10, -15, 20, -25
   • Identified Pattern: Multiples of 5, alternating signs
   • General Term: a_n = (-1)^n * 5n
4. Sequence: 1, 4, 9, 16, 25
   • Identified Pattern: Perfect squares
   • General Term: a_n = n^2
5. Sequence: 3, -6, 9, -12, 15
   • Identified Pattern: Multiples of 3, alternating signs
   • General Term: a_n = (-1)^n * 3n

Example Calculation

• Given General Term: a_n = (n - 3)^n
  • Calculate first 5 terms:
    1. a_1 = (1 - 3)^1 = -2^1 = -2
    2. a_2 = (2 - 3)^2 = -1^2 = 1
    3. a_3 = (3 - 3)^3 = 0^3 = 0
    4. a_4 = (4 - 3)^4 = 1^4 = 1
    5. a_5 = (5 - 3)^5 = 2^5 = 32

Summary

• Understanding sequences involves identifying patterns and formulating general terms that describe the sequence.
• Different sequences follow different rules (e.g., arithmetic, geometric, cumulative, etc.).

Additional Practice

• Finding the general term for more complex sequences helps in understanding the underlying mathematical principles.