Notes on Arithmetic Sequences

Definition of Arithmetic Sequence

• An arithmetic sequence is a sequence where the difference between consecutive numbers is constant.

Key Components

• TN: Term value (e.g., 11 is the second term when n=2).
• a: First term in the sequence.
• D: Common difference (the constant added to each term).
• Negative D: If the sequence decreases, D can be a negative number.

Example Calculation

• Given the sequence: 5, 11, 17...
• Here:
  • a = 5
  • D = 6
• Formula to find the nth term:
  • TN = a + (n - 1)D
  • Example for n=2:
    • TN = 5 + (2 - 1) * 6 = 5 + 6 = 11*

Finding the Formula for a Given Sequence

1. Identify the first term (a) and the common difference (D).
2. Plug values into the formula:
   • Example sequence: 9, 13, 17...
   • a = 9, D = 4.
3. Formula Development:
   • TN = a + (n - 1)D
   • TN = 9 + (n - 1)4
   • After simplification: TN = 4n + 5
4. Finding a Specific Term:
   • For n=12: TN = 4(12) + 5 = 53.

Recursive Formula Example

• Given first term = 5, second term = 11:
  • Find third term (T3) using the recursion formula:
    • T3 = T2 + 6 = 11 + 6 = 17.
• Confirming it’s arithmetic allows the use of the explicit formula.

Determining the Number of Terms

• Given part of a sequence and the last term, find how many terms exist.
  • Example: First term = -3, difference = 5, last term = 152.
  • Use explicit formula: TN = a + (n - 1)D
  • Solve for n:
    • 152 = -3 + (n - 1)5
    • Solve the equation to find n.

Given Two Random Terms

• Given n and TN values, set up two equations to solve for a and D:
  1. Example: 7th term = 36
     • a + 6D = 36
  2. Example: 15th term = 68
     • a + 14D = 68
• Use elimination or substitution to find values of a and D.
  • Once found, form the explicit equation for the sequence: TN = a + (n - 1)D.