13.3 Geometric Sequences

Geometric sequences are mathematical patterns where each term is found by multiplying the previous term by a fixed number called the common ratio. These sequences are powerful tools for modeling growth or decay in various real-world scenarios.

Understanding geometric sequences opens doors to exponential functions and compound interest calculations. By mastering the common ratio, term generation, and formulas, you'll gain insights into patterns that appear in finance, biology, and physics.

Geometric Sequences

Common ratio in geometric sequences

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• Sequence of numbers where each term is found by multiplying the previous term by a fixed, non-zero number called the common ratio ($r$)
• Find the common ratio by dividing any term in the sequence by the previous term
  • Formula: $r = \frac{a_{n+1}}{[a_n](https://www.fiveableKeyTerm:a_n)}$, where $a_n$ is the $n$th term of the sequence
• Common ratio can be positive (2, 4, 8), negative (-3, 6, -12), or a fraction (1, $\frac{1}{2}$, $\frac{1}{4}$)
  • If $|r| > 1$, terms increase in absolute value as $n$ increases (2, 6, 18)
  • If $|r| < 1$, terms decrease in absolute value as $n$ increases (80, 40, 20)

Term generation for geometric sequences

• Generate terms by starting with the first term $a_1$ and multiplying by the common ratio $r$ for each subsequent term
  • $a_2 = a_1 \cdot r$
  • $a_3 = a_2 \cdot r = a_1 \cdot r^2$
  • $a_4 = a_3 \cdot r = a_1 \cdot r^3$
• General formula for the $n$th term: $a_n = a_1 \cdot r^{n-1}$
• Examples:
  • Sequence: 3, 6, 12, 24; $a_1 = 3$, $r = 2$
  • Sequence: 1000, 100, 10, 1; $a_1 = 1000$, $r = \frac{1}{10}$

Recursive formulas for sequence analysis

• Recursive formula defines each term in relation to the previous term
  • Formula: $a_n = a_{n-1} \cdot r$, where $a_1$ is given
• Extend a sequence using a recursive formula by multiplying the previous term by the common ratio to find the next term
  • Example: $a_1 = 5$, $r = 3$; $a_2 = 5 \cdot 3 = 15$, $a_3 = 15 \cdot 3 = 45$
• Useful for generating terms step-by-step and analyzing patterns in the sequence

Explicit formulas for specific terms

• Explicit formula allows finding the $n$th term directly without calculating previous terms
  • Formula: $a_n = a_1 \cdot r^{n-1}$, where $a_1$ is the first term, $r$ is the common ratio, and $n$ is the position of the term
• Find a specific term by substituting values for $a_1$, $r$, and $n$ into the formula and simplify
  • Example: $a_1 = 2$, $r = 4$, find $a_5$
    1. $a_5 = 2 \cdot 4^{5-1}$
    2. $a_5 = 2 \cdot 4^4$
    3. $a_5 = 2 \cdot 256 = 512$
• Useful for finding terms far into the sequence without calculating each preceding term ($a_{50}$, $a_{100}$)

Connections to Exponential Functions and Applications

• Geometric sequences are closely related to exponential functions, where the common ratio serves as the base
• The explicit formula for a geometric sequence ($a_n = a_1 \cdot r^{n-1}$) is a discrete version of an exponential function
• Exponents in geometric sequences represent repeated multiplication by the common ratio
• Logarithms can be used to solve equations involving geometric sequences
• Real-world applications include modeling compound interest in financial calculations