Geometric Sequences and Their General Term

Introduction

Key Concepts

Definition of Geometric Sequences

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a constant non-zero number called the common ratio ($r$). Mathematically, if the first term is $a_1$, the sequence can be represented as:

$$ a_1, a_1 \cdot r, a_1 \cdot r^2, a_1 \cdot r^3, \ldots $$

For example, consider the sequence 2, 6, 18, 54, ... Here, each term is obtained by multiplying the previous term by 3, making 3 the common ratio.

General Term of a Geometric Sequence

The general term or the $n^{th}$ term of a geometric sequence provides a way to find any term in the sequence without listing all preceding terms. It is given by the formula:

$$ a_n = a_1 \cdot r^{n-1} $$

Where:

• $a_n$ is the $n^{th}$ term.
• $a_1$ is the first term.
• $r$ is the common ratio.
• $n$ is the term number.

Using the previous example with $a_1 = 2$ and $r = 3$, the $5^{th}$ term ($a_5$) would be:

$$ a_5 = 2 \cdot 3^{5-1} = 2 \cdot 81 = 162 $$

Sum of the First $n$ Terms

The sum of the first $n$ terms of a geometric sequence ($S_n$) can be calculated using the formula:

$$ S_n = a_1 \cdot \frac{1 - r^n}{1 - r}, \quad \text{for } r \neq 1 $$

For example, to find the sum of the first 4 terms of the sequence 2, 6, 18, 54:

$$ S_4 = 2 \cdot \frac{1 - 3^4}{1 - 3} = 2 \cdot \frac{1 - 81}{-2} = 2 \cdot 40 = 80 $$

Properties of Geometric Sequences

• Common Ratio ($r$): Determines the rate at which the sequence increases or decreases. If $|r| > 1$, the sequence grows; if $0 < |r| < 1$, it decays.
• Exponential Nature: Geometric sequences exhibit exponential growth or decay, making them suitable for modeling population growth, radioactive decay, and interest calculations.
• Non-Linearity: Unlike arithmetic sequences, geometric sequences are not linear due to their multiplicative progression.

Examples of Geometric Sequences

Example 1: Find the $6^{th}$ term of a geometric sequence where the first term $a_1 = 5$ and the common ratio $r = 2$.

Using the general term formula: $$ a_6 = 5 \cdot 2^{6-1} = 5 \cdot 32 = 160 $$

Example 2: Determine the sum of the first 5 terms of the geometric sequence 3, 12, 48, ...

First, identify $a_1 = 3$ and $r = \frac{12}{3} = 4$. Then, apply the sum formula: $$ S_5 = 3 \cdot \frac{1 - 4^5}{1 - 4} = 3 \cdot \frac{1 - 1024}{-3} = 3 \cdot \frac{-1023}{-3} = 3 \cdot 341 = 1023 $$>

Applications of Geometric Sequences

Geometric sequences are widely applicable in various fields:

• Finance: Calculating compound interest where each period's interest is applied to the accumulated amount.
• Biology: Modeling population growth where each generation multiplies by a constant factor.
• Physics: Describing radioactive decay processes.
• Computer Science: Analyzing algorithms with exponential time complexities.

Convergence and Divergence

The behavior of a geometric sequence as $n$ approaches infinity depends on the common ratio $r$:

• If |r| < 1: The sequence converges to 0.
• If |r| ≥ 1: The sequence diverges, meaning it grows without bound.

For instance, the sequence with $a_1 = 1$ and $r = \frac{1}{2}$ converges as its terms approach zero.

Identifying Geometric Sequences

To determine whether a given sequence is geometric:

1. Calculate the ratio of consecutive terms.
2. If the ratio remains constant, the sequence is geometric.
3. If the ratio varies, the sequence is not geometric.

Example: Determine if the sequence 5, 15, 45, 135, ... is geometric.

Calculate the ratios: $$ \frac{15}{5} = 3, \quad \frac{45}{15} = 3, \quad \frac{135}{45} = 3 $$ Since the ratio is constant ($r = 3$), the sequence is geometric.

Recursive Definition of Geometric Sequences

Beyond the explicit formula, geometric sequences can also be defined recursively:

$$ a_1 = \text{initial term}, \quad a_{n} = a_{n-1} \cdot r \quad \text{for } n > 1 $$>

This recursive approach emphasizes the relationship between consecutive terms.

Geometric Mean

The geometric mean between two positive numbers $a$ and $b$ is defined as:

$$ \sqrt{a \cdot b} $$>

In a geometric sequence, each term is the geometric mean of its neighboring terms.

Common Scenarios and Problem-Solving Techniques

When tackling problems involving geometric sequences, consider the following strategies:

• Identifying Parameters: Clearly determine $a_1$, $r$, and the term number $n$.
• Using Logarithms: For problems involving exponential growth or decay, logarithms can be useful in solving for unknowns.
• Graphical Representation: Plotting terms can provide insights into the behavior of the sequence.

Example: A population of bacteria doubles every hour. If there are initially 500 bacteria, how many bacteria will there be after 8 hours?

Here, $a_1 = 500$, $r = 2$, and $n = 8$. Using the general term: $$ a_8 = 500 \cdot 2^{8-1} = 500 \cdot 128 = 64,000 $$>

Common Mistakes to Avoid

• Confusing the common ratio with the common difference used in arithmetic sequences.
• Incorrectly applying the general term formula, especially the exponent.
• Neglecting to verify whether a sequence is truly geometric before applying formulas.
• Handling negative ratios without considering their impact on the sequence's direction.

Comparison Table

Aspect	Arithmetic Sequences	Geometric Sequences
Definition	Each term is obtained by adding a constant difference ($d$).	Each term is obtained by multiplying by a constant ratio ($r$).
General Term	$a_n = a_1 + (n-1) \cdot d$	$a_n = a_1 \cdot r^{n-1}$
Sum of Terms	$S_n = \frac{n}{2} [2a_1 + (n-1)d]$	$S_n = a_1 \cdot \frac{1 - r^n}{1 - r}$
Growth Type	Linear growth or decline.	Exponential growth or decay.
Applications	Salary increments, simple interest.	Compound interest, population growth.
Graph Shape	Straight line.	Exponential curve.

Summary and Key Takeaways