The Main Idea

• Definition: A geometric sequence is a sequence where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.
• General Form: [latex]{a_n} = {a_1, a_1r, a_1r^2, a_1r^3, ..., a_1r^{n-1}}[/latex]
• Common Ratio: [latex]r = \frac{a_n}{a_{n-1}}[/latex] for any [latex]n \geq 2[/latex]
• Explicit Formula: [latex]a_n = a_1r^{n-1}[/latex]
• Key Properties:
  • Can model exponential growth or decay
  • Terms can be positive, negative, or alternating
  • The absolute values of terms either strictly increase, strictly decrease, or remain constant
• Real-world Applications:
  • Compound interest
  • Population growth
  • Radioactive decay
  • Moore’s Law in computing

The Main Idea

• Term Generation Formula: [latex]a_n = a_1 \cdot r^{n-1}[/latex] Where:
  • [latex]a_n[/latex] is the [latex]n[/latex]th term
  • [latex]a_1[/latex] is the first term
  • [latex]r[/latex] is the common ratio
  • [latex]n[/latex] is the term number
• Recursive Process: Each term is the product of the previous term and the common ratio: [latex]a_n = a_{n-1} \cdot r[/latex]
• Pattern Recognition:
  • If [latex]|r| > 1[/latex], the absolute values of terms increase
  • If [latex]0 < |r| < 1[/latex], the absolute values of terms decrease
  • If [latex]r < 0[/latex], the terms alternate in sign
• Exponential Nature: The exponent of r in each term is one less than the term’s position

The Main Idea

• Geometric Sequence as an Exponential Function:
  • A geometric sequence is an exponential function with a domain of positive integers.
  • The common ratio (r) is the base of the function.
• Explicit Formula:
  • General form: [latex]a_n = a_1r^{n-1}[/latex]
  • [latex]a_n[/latex]: nth term of the sequence
  • [latex]a_1[/latex]: first term of the sequence
  • [latex]r[/latex]: common ratio
  • [latex]n[/latex]: term number

The Main Idea

• Recursive Formula Definition:
  • Allows finding any term using the previous term
  • Each term is the product of the common ratio and the previous term
  • Initial term must be given
• General Recursive Formula:
  • [latex]a_n = ra_{n-1}, n \ge 2[/latex]
  • [latex]r[/latex]: common ratio
  • [latex]a_n[/latex]: nth term
  • [latex]a_{n-1}[/latex]: previous term
  • [latex]a_1[/latex]: first term (must be specified)