12.3 Geometric Sequences and Series

Geometric sequences are number patterns where each term is found by multiplying the previous term by a fixed number called the common ratio. They're useful for modeling exponential growth or decay in various real-world scenarios.

Understanding geometric sequences helps you grasp compound interest, population growth, and depreciation. You'll learn to identify these sequences, calculate terms, find sums of finite and infinite series, and apply them to practical problems.

Geometric Sequences

Geometric sequences identification

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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 $r$ (2, 6, 18, 54, ...)
• General form: $a_n = a_1 \cdot r^{n-1}$, where $a_1$ is the first term, $r$ is the common ratio, and $n$ is the term number
• Characteristics:
  • Ratio between any two consecutive terms is constant (common ratio)
  • Terms increase exponentially if $|r| > 1$ (2, 8, 32, 128, ...) or decrease exponentially if $0 < |r| < 1$ (1, $\frac{1}{2}$, $\frac{1}{4}$, $\frac{1}{8}$, ...)
  • Sequence alternates between positive and negative if $r < 0$ (1, -2, 4, -8, ...)
• Can be defined recursively, where each term is expressed in terms of the previous term

General term calculation

• Formula for the nth term (general term) of a geometric sequence: $a_n = a_1 \cdot r^{n-1}$
  • $a_n$ represents the nth term
  • $a_1$ is the first term
  • $r$ is the common ratio
  • $n$ is the term number
• Find the common ratio by dividing any term by the previous term: $r = \frac{a_{n+1}}{a_n}$
• Example: Find the 7th term of the geometric sequence 4, 12, 36, 108, ...
  1. Identify $a_1 = 4$ and calculate $r = \frac{12}{4} = 3$
  2. Use the general term formula with $n = 7$: $a_7 = 4 \cdot 3^{7-1} = 4 \cdot 3^6 = 2,916$
• The geometric mean of two terms in a sequence is the middle term between them

Finite sequence sums

• Formula for the sum of a finite geometric sequence (series) with $n$ terms: $S_n = \frac{a_1(1-[r^n](https://www.fiveableKeyTerm:r^n))}{1-r}$
  • $S_n$ is the sum of the first $n$ terms
  • $a_1$ is the first term
  • $r$ is the common ratio
  • $n$ is the number of terms
• Example: Find the sum of the first 5 terms of the geometric sequence 3, 9, 27, 81, ...
  1. Identify $a_1 = 3$, $r = 3$, and $n = 5$
  2. Use the sum formula: $S_5 = \frac{3(1-3^5)}{1-3} = \frac{3(1-243)}{-2} = 363$

Infinite series evaluation

• An infinite geometric series converges (has a finite sum) if $|r| < 1$
• Formula for the sum of an infinite geometric series: $S_\infty = \frac{a_1}{1-r}$
  • $S_\infty$ is the sum of the infinite series
  • $a_1$ is the first term
  • $r$ is the common ratio (must be between -1 and 1, exclusive)
• Example: Find the sum of the infinite geometric series 5, 2, 0.8, 0.32, ...
  1. Identify $a_1 = 5$ and $r = \frac{2}{5} = 0.4$
  2. Use the infinite sum formula: $S_\infty = \frac{5}{1-0.4} = \frac{5}{0.6} = \frac{25}{3} \approx 8.33$
• The limit of a geometric sequence as n approaches infinity can be found using the ratio test

Real-world applications

1. Compound interest: $A = P(1+r)^n$
   • $A$ is the final amount
   • $P$ is the principal (initial investment)
   • $r$ is the interest rate per compounding period (expressed as a decimal)
   • $n$ is the number of compounding periods
2. Population growth: $P_n = P_0(1+r)^n$
   • $P_n$ is the population after $n$ periods
   • $P_0$ is the initial population
   • $r$ is the growth rate per period (expressed as a decimal)
3. Depreciation: $V_n = V_0(1-r)^n$
   • $V_n$ is the value after $n$ periods
   • $V_0$ is the initial value
   • $r$ is the depreciation rate per period (expressed as a decimal)

Related Concepts

• Arithmetic sequences: Similar to geometric sequences, but with a constant difference between terms instead of a constant ratio
• Recursive definition: A way to define sequences where each term is expressed in terms of previous terms
• Ratio test: A method used to determine the convergence of infinite series, including geometric series