The Main Idea

• Definition:
  • A sequence where the difference between consecutive terms is constant
  • This constant difference is called the common difference, denoted ‘[latex]d[/latex]‘
• General Form: [latex]{a_n} = {a_1, a_1+d, a_1+2d, a_1+3d, ...}[/latex]
• Key Properties:
  • Linear growth
  • Graphed as points on a straight line
• Real-world Applications:
  • Depreciation (straight-line method)
  • Linear salary increases
  • Arithmetic interest

The Main Idea

• General Term Formula: [latex]a_n = a_1 + (n - 1)d[/latex] Where:
  • [latex]a_n[/latex] is the nth term
  • [latex]a_1[/latex] is the first term
  • [latex]n[/latex] is the term number
  • [latex]d[/latex] is the common difference
• Pattern Recognition:
  • The coefficient of [latex]d[/latex] is always one less than the term number
• Reverse Engineering:
  • Given any two terms, we can find the common difference and first term

The Main Idea

• Explicit Formula: [latex]a_n = a_1 + d(n - 1)[/latex] Where:
  • [latex]a_n[/latex] is the nth term
  • [latex]a_1[/latex] is the first term
  • [latex]d[/latex] is the common difference
  • [latex]n[/latex] is the term number
• Deriving the Formula:
  • Observe the pattern: [latex]a_1, a_1 + d, a_1 + 2d, a_1 + 3d, ...[/latex]
  • Generalize to [latex]a_1 + (n-1)d[/latex] for the nth term
• Applications:
  • Quickly find any term without calculating all previous terms
  • Analyze long-term behavior of sequences
• Connection to Linear Functions:
  • The explicit formula represents a linear function when n is treated as a continuous variable

The Main Idea

• Recursive Formula: [latex]a_n = a_{n-1} + d, \text{ for } n \geq 2[/latex] Where:
  • [latex]a_n[/latex] is the nth term
  • [latex]a_{n-1}[/latex] is the previous term
  • [latex]d[/latex] is the common difference
  • Initial condition: [latex]a_1[/latex] (first term) must be specified
• Characteristics:
  • Defines each term based on the previous term
  • Requires knowing the first term and common difference
• Advantages:
  • Often mirrors the natural way sequences are generated
  • Can be easier to understand for some types of sequences
• Limitations:
  • Can be computationally intensive for finding terms far in the sequence
  • May not provide immediate insight into the long-term behavior

The Main Idea

• Concept:
  • In a finite arithmetic sequence, we can determine the total number of terms if we know the first term, last term, and common difference.
• Key Formula: [latex]a_n = a_1 + d(n - 1)[/latex] Where:
  • [latex]a_n[/latex] is the last term
  • [latex]a_1[/latex] is the first term
  • [latex]d[/latex] is the common difference
  • [latex]n[/latex] is the number of terms (what we’re solving for)
• Process:
  • Find the common difference
  • Set up the equation using the first and last terms
  • Solve for [latex]n[/latex]
• Interpretation:
  • The solution [latex]n[/latex] must be a positive integer
  • If [latex]n[/latex] is not a whole number, round up to the next integer for the minimum number of terms needed