Notes on Arithmetic Sequence

Definition

• An arithmetic sequence is a number pattern where the difference between any two consecutive numbers is constant.
  • Example: Difference between 23 and 25 is 2; difference between 13 and 15 is also 2.

Key Components of Arithmetic Sequences

• A: Term number 1 (first term)
• N: Position number (where the term is located in the sequence)
  • Example: 9 is at position 1; 19 is at position 6.
• D: Common difference between terms (e.g., in this case, D = 2)
• Tn: Value of the term at position N.
  • Example: 25 is at position 9 and has a value of 25.

Formula for Arithmetic Sequences

• The general formula is:
  Tn = A + (N - 1) * D
  Where:
  • Tn is the term value at position N
  • A is the first term
  • N is the position number
  • D is the common difference*

Working with the Formula

• Example 1: Finding a specific term

  • If we want to find the term at position 7:
    • Given: A = 9, D = 2
    • Use the formula:
      • T7 = A + (7 - 1) * D
      • T7 = 9 + (6 * 2)
      • T7 = 9 + 12 = 21

• Example 2: Finding the position of a known term

  • If we know a term value (e.g., 17) but not its position:
    • Set up the equation:
      • 17 = A + (N - 1) * D
      • 17 = 9 + (N - 1) * 2
      • Rearranging gives:
        • 8 = (N - 1) * 2
        • 4 = N - 1
        • N = 5
    • Confirming: 17 is indeed at position 5.*

Summary

• Key Takeaway: Use the formula Tn = A + (N - 1) * D to determine values or positions in an arithmetic sequence.*