3.11 Geometric Sequences

Geometric sequences are like mathematical dominos, each number falling into place by multiplying the previous one by a fixed ratio. They're everywhere, from compound interest to population growth, showing how small changes can lead to big results over time.

Understanding geometric sequences unlocks powerful tools for predicting future values and calculating sums. Whether finite or infinite, these sequences help us model real-world scenarios where consistent growth or decay occurs, making complex patterns more manageable.

Geometric Sequences

Geometric sequences and common ratios

• A sequence of numbers where each term is found by multiplying the previous term by a fixed, non-zero number called the common ratio
• To find the common ratio, divide any term by the previous term and the ratio will be the same for any pair of consecutive terms (2, 6, 18, 54, ... has a common ratio of 3)
• Formula for the common ratio: $r = \frac{a_{n+1}}{a_n}$
  • $a_n$ is the $n$th term of the sequence
• The recursive definition of a geometric sequence is $a_n = r \cdot a_{n-1}$, where $a_1$ is the first term

Formula for nth term

• The $n$th term of a geometric sequence can be found using the formula: $a_n = a_1 \cdot r^{n-1}$
  • $a_n$ is the $n$th term
  • $a_1$ is the first term
  • $r$ is the common ratio
  • $n$ is the position of the term
• To find the $n$th term, substitute the values for $a_1$, $r$, and $n$ into the formula and simplify
• In the sequence 2, 6, 18, 54, ..., to find the 10th term:
  1. Identify $a_1 = 2$, $r = 3$, $n = 10$
  2. Substitute into the formula: $a_{10} = 2 \cdot 3^{10-1} = 2 \cdot 3^9$
  3. Simplify: $a_{10} = 39,366$

Sums of finite geometric sequences

• The sum of a finite geometric sequence can be found using the formula: $S_n = \frac{a_1(1-r^n)}{1-r}$
  • $S_n$ is the sum of the first $n$ terms (also called the partial sum)
  • $a_1$ is the first term
  • $r$ is the common ratio
  • $n$ is the number of terms
• To find the sum, substitute the values for $a_1$, $r$, and $n$ into the formula and simplify
• Real-world applications include:
  • Compound interest: balance of an account earning compound interest grows by a fixed percentage each period, forming a geometric sequence
  • Population growth: if a population grows by a fixed percentage each year, the population values form a geometric sequence (bacteria doubling every hour)
  • Depreciation: value of an asset that depreciates by a fixed percentage each year forms a geometric sequence (car losing 10% of its value annually)

Infinite Geometric Sequences and Series

• A geometric sequence can be infinite if it continues indefinitely
• An infinite geometric sequence can be convergent or divergent
  • A convergent sequence approaches a finite limit as n approaches infinity
  • A divergent sequence (such as 2, 4, 8, 16, ...) does not approach a finite limit
• The geometric series is the sum of terms in a geometric sequence
• For an infinite geometric series with |r| < 1, the sum can be calculated using the formula: $S_{\infty} = \frac{a_1}{1-r}$
  • This formula represents the limit of the partial sums as n approaches infinity