Sequence, Finite, and Infinite Sequence

Key Concepts

• Sequence: Function whose domain is the set of positive integers. Represents an ordered list of numbers.
• Finite Sequence: Ends with a specific last term.
• Infinite Sequence: Continues indefinitely, often represented with ellipsis (...).
• Term: Each number in a sequence, denoted by the order. For example, 5, 15, 25, etc., are the first, second, and third terms.
• General Term Rule: Mathematical expression for generating terms in a sequence.

Pattern Searching in Sequences

Examples of Pattern Identification

1. Letters Sequence:

   • Given: A, B, G, J
   • Pattern: Skip two letters
   • Next Terms: M, P, S

2. Odd Numbers Sequence:

   • Given: 1, 3, 5, 7
   • Pattern: Consecutive odd numbers
   • Next Terms: 9, 11, 13

3. Perfect Squares Sequence:

   • Given: 1, 4, 9, 16, 25
   • Pattern: Perfect squares
   • Next Terms: 36, 49, 64

4. Add Ten Sequence:

   • Given: 5, 15, 25, 35
   • Pattern: Add 10
   • Next Terms: 45, 55, 65

5. Incremented Sum Sequence:

   • Given: 1, 3, 6, 10, 15, 21, 28
   • Pattern: 1 + 2, 3 + 3, 6 + 4, etc.
   • Next Terms: 36, 45

Understanding Sequences

Definitions

• Sequence: A function with positive integers as its domain resulting in an ordered list of numbers.
• Finite Sequence: Has a specific last term, denoted often as n.
• Infinite Sequence: Does not have a specific last term, often indicated by ....
• Terms: Each number in a sequence, identified by placement (1st term, 2nd term, etc.).

Denotation of Terms

• First Term (a1): Example – 1
• Second Term (a2): Example – 3
• Third Term (a3): Example – 6

General Term and Proof

Example of General Term Calculation

• Rule: n/2 * (n + 1)

• Example for 1st Term:

  • Substitute n = 1
  • Calculation: (1/2)*(1+1) = 1

• Example for 3rd Term:

  • Substitute n = 3
  • Calculation: (3/2)(3+1) = 6

Example of General Term Derivation

• Given Numbers: 1, 4, 9, 16, 25
• Pattern: Perfect squares
• General Term: n^2

Finding General Terms from Sequences

Given Sequences and Their General Terms

1. 1, 8, 27, 64, 125:

   • General Term: n^3

2. 1/1, 1/2, 1/3, 1/4, 1/5:

   • General Term: 1/n

3. -5, 10, -15, 20, -25:

   • General Term: (-1)^n * 5n

4. 1, 4, 9, 16, 25:

   • General Term: n^2

5. 3, -6, 9, -12, 15:

   • General Term: (-1)^n * 3n

Concluding Notes

• Remember to check the pattern carefully when deriving the general term.
• Always verify the derived sequence through substitution and calculation to confirm accuracy.