Geometric Sequences and Their General Term

Introduction

Key Concepts

Understanding 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 called the common ratio. This ratio is denoted by \( r \) and is a key factor in determining the behavior of the sequence.

Definition: A geometric sequence can be defined as: $$ a_n = a_1 \times r^{(n-1)} $$ where:

• \( a_n \) is the \( n^{th} \) term of the sequence.
• \( a_1 \) is the first term.
• \( r \) is the common ratio.
• \( n \) is the term number.

Identifying the Common Ratio

The common ratio \( r \) is pivotal in geometric sequences. It can be determined by dividing any term in the sequence by the preceding term: $$ r = \frac{a_{n}}{a_{n-1}} $$ For example, in the sequence 3, 6, 12, 24, ..., the common ratio \( r = 2 \).

General Term of a Geometric Sequence

The general term formula allows the calculation of any term in the sequence without needing to know all the previous terms. It is given by: $$ a_n = a_1 \times r^{(n-1)} $$ This formula is essential for finding specific terms and understanding the sequence's progression.

Sum of a Geometric Series

When considering the sum of terms in a geometric sequence, we refer to it as a geometric series. The sum of the first \( n \) terms \( S_n \) is calculated as: $$ S_n = a_1 \times \frac{1 - r^n}{1 - r} \quad \text{for} \quad r \neq 1 $$ If \( |r| < 1 \), the series converges, and its sum approaches: $$ S = \frac{a_1}{1 - r} $$

Applications of Geometric Sequences

Geometric sequences are widely applicable in various fields such as finance (compound interest), biology (population growth), physics (radioactive decay), and computer science (algorithm complexity). Understanding their properties enables the modeling and solving of real-world problems.

Examples and Problem Solving

Consider the sequence: 5, 15, 45, 135, ...

• First Term (\( a_1 \)): 5
• Common Ratio (\( r \)): \( \frac{15}{5} = 3 \)
• General Term (\( a_n \)): \( 5 \times 3^{(n-1)} \)

To find the 5th term (\( a_5 \)): $$ a_5 = 5 \times 3^{(5-1)} = 5 \times 81 = 405 $$

Properties of Geometric Sequences

• Exponential Growth or Decay: Depending on the value of \( r \), the sequence can represent exponential growth (\( r > 1 \)) or decay (\( 0 < r < 1 \)).
• Non-linear Progression: Unlike arithmetic sequences, geometric sequences do not increase or decrease by a constant difference but by a constant ratio, leading to non-linear progression.
• Infinite Sequences: Geometric sequences can be extended infinitely, especially when modeling processes that continue indefinitely.

Graphical Representation

Plotting geometric sequences on a graph typically results in an exponential curve. For \( r > 1 \), the graph rises sharply, indicating rapid growth. For \( 0 < r < 1 \), the graph approaches zero, showcasing decay.

Real-World Example: Compound Interest

In finance, compound interest calculations utilize geometric sequences. The amount \( A \) after \( n \) periods is given by: $$ A = P \times \left(1 + \frac{r}{100}\right)^n $$ where:

• \( P \) is the principal amount.
• \( r \) is the annual interest rate.
• \( n \) is the number of periods.

This formula exemplifies a geometric sequence where each term represents the investment's growth over time.

Error Checking and Fact Verification

Ensuring the accuracy of calculations in geometric sequences is crucial. Miscalculations in determining the common ratio or applying the general term formula can lead to incorrect results. Always cross-verify results by substituting back into the sequence.

Advanced Concepts

Theoretical Foundations of Geometric Sequences

Delving deeper, geometric sequences are intrinsically linked to exponential functions. The general term formula: $$ a_n = a_1 \times r^{(n-1)} $$ can be viewed as a discrete version of the continuous exponential function: $$ f(x) = a_1 \times e^{kx} $$ where \( k \) relates to the growth or decay rate. This connection is pivotal in calculus, aiding in understanding limits and derivatives of exponential functions.

Mathematical Derivations and Proofs

Deriving the sum of a geometric series involves understanding infinite series and convergence criteria. For a geometric series to converge, the absolute value of the common ratio must be less than one (\( |r| < 1 \)). The derivation starts with the sum formula: $$ S_n = a_1 + a_1r + a_1r^2 + \cdots + a_1r^{(n-1)} $$ Multiplying both sides by \( r \): $$ rS_n = a_1r + a_1r^2 + \cdots + a_1r^{(n)} $$ Subtracting the two equations: $$ S_n - rS_n = a_1 - a_1r^n $$ Solving for \( S_n \): $$ S_n = a_1 \times \frac{1 - r^n}{1 - r} $$ As \( n \) approaches infinity and \( |r| < 1 \), \( r^n \) approaches zero, yielding: $$ S = \frac{a_1}{1 - r} $$

Complex Problem-Solving

Consider a scenario where a bacteria culture doubles every hour. If the initial population is 500 bacteria, determine the population after 10 hours.

• First Term (\( a_1 \)): 500
• Common Ratio (\( r \)): 2
• General Term (\( a_{10} \)): $$ a_{10} = 500 \times 2^{(10-1)} = 500 \times 512 = 256,000 $$

Thus, after 10 hours, the population will be 256,000 bacteria.

Interdisciplinary Connections

Geometric sequences intersect with various disciplines:

• Physics: Radioactive decay follows a geometric sequence as the number of undecayed nuclei decreases by a constant ratio over time.
• Economics: Compound interest calculations in finance use geometric sequences to model investment growth.
• Biology: Population models often assume geometric growth or decay under ideal conditions.
• Computer Science: Analyzing algorithmic complexity, especially in recursive algorithms, involves geometric sequences.

Understanding these connections enhances the application of geometric sequences beyond pure mathematics.

Infinite Geometric Series

An infinite geometric series extends indefinitely. The sum \( S \) of an infinite geometric series is given by: $$ S = \frac{a_1}{1 - r} \quad \text{for} \quad |r| < 1 $$ This concept is crucial in calculus and mathematical analysis, particularly in power series and convergence tests.

Applications in Calculus

Geometric sequences serve as building blocks for power series, which are used to represent functions in calculus. For instance, the exponential function can be expressed as an infinite geometric series: $$ e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} $$ Though not a geometric series in the strictest sense, this representation relies on the principles of geometric progression.

Exploring Limits and Convergence

Understanding the behavior of geometric sequences as \( n \) approaches infinity involves limit calculations: $$ \lim_{n \to \infty} a_n = \lim_{n \to \infty} a_1 \times r^{(n-1)} $$ - If \( |r| < 1 \), the limit is 0. - If \( |r| = 1 \), the sequence does not converge. - If \( |r| > 1 \), the sequence diverges to infinity or negative infinity.

Advanced Problem Example

Problem: A car depreciates in value by 15% each year. If the car's initial value is \$20,000, what will be its value after 7 years?

• First Term (\( a_1 \)): \$20,000
• Common Ratio (\( r \)): \( 1 - 0.15 = 0.85 \)
• General Term (\( a_7 \)): $$ a_7 = 20000 \times 0.85^{(7-1)} = 20000 \times 0.85^{6} \approx 20000 \times 0.3771495 \approx \$7,543 $$

After 7 years, the car's value will be approximately \$7,543.

Exploring Recursive Definitions

While the general term provides a direct formula to find any term, geometric sequences can also be defined recursively: $$ a_1 = \text{given} $$ $$ a_{n} = a_{n-1} \times r \quad \text{for} \quad n > 1 $$ This recursive approach is beneficial in algorithm design and understanding iterative processes.

Connection to Compound Interest Formulas

The compound interest formula is a direct application of geometric sequences: $$ A = P \times \left(1 + \frac{r}{n}\right)^{nt} $$ where:

• \( A \) is the amount of money accumulated after \( t \) years.
• \( P \) is the principal investment amount.
• \( r \) is the annual interest rate.
• \( n \) is the number of times that interest is compounded per year.

Each compounding period represents a term in a geometric sequence, with the common ratio being \( \left(1 + \frac{r}{n}\right) \).

Advanced Graphing Techniques

Graphing geometric sequences involves plotting discrete points that often form an exponential curve. Advanced techniques include:

• Logarithmic Scaling: Applying logarithmic scales can linearize exponential growth or decay, making trends easier to analyze.
• Asymptotic Behavior: Understanding asymptotes helps in identifying the limits that the sequence approaches.

Exploring Growth and Decay Rates

Growth and decay rates in geometric sequences are determined by the common ratio \( r \):

• Growth: If \( r > 1 \), the sequence exhibits exponential growth.
• Decay: If \( 0 < r < 1 \), the sequence demonstrates exponential decay.

Analyzing these rates is crucial in fields like epidemiology, finance, and environmental science.

Comparison Table

Aspect	Arithmetic Sequence	Geometric Sequence
Definition	Each term is obtained by adding a constant difference.	Each term is obtained by multiplying by a constant ratio.
General Term	$$ a_n = a_1 + (n-1)d $$	$$ a_n = a_1 \times r^{(n-1)} $$
Graph	Linear progression.	Exponential curve.
Sum of Terms	$$ S_n = \frac{n}{2} \times (2a_1 + (n-1)d) $$	$$ S_n = a_1 \times \frac{1 - r^n}{1 - r} \quad \text{for} \quad r \neq 1 $$
Applications	Salary increments, loan repayments.	Population growth, compound interest.
Growth Rate	Constant difference.	Constant ratio.

Summary and Key Takeaways