Definition and General Term of Geometric Sequences

Introduction

Key Concepts

What is a Geometric Sequence?

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the preceding term by a constant called the common ratio. This constant ratio differentiates geometric sequences from other types of sequences, such as arithmetic sequences, where the difference between consecutive terms is constant.

General Term of a Geometric Sequence

The general term of a geometric sequence allows us to find any term in the sequence without listing all the previous terms. It is expressed using the formula:

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

Here:

• $a_n$ represents the nth term of the sequence.
• $a_1$ is the first term.
• $r$ is the common ratio.
• $n$ is the term number.

For example, if the first term ($a_1$) is 3 and the common ratio ($r$) is 2, the sequence proceeds as 3, 6, 12, 24, and so on. The fifth term ($a_5$) can be calculated as:

$$a_5 = 3 \times 2^{5-1} = 3 \times 16 = 48$$

Derivation of the General Term Formula

To derive the general term formula, consider the sequence:

• $a_1$
• $a_2 = a_1 \times r$
• $a_3 = a_2 \times r = a_1 \times r^2$
• ...
• $a_n = a_1 \times r^{n-1}$

By observing the pattern, it becomes clear that each term is the product of the first term and the common ratio raised to the power of one less than the term number.

Properties of Geometric Sequences

• Common Ratio ($r$): Indicates how each term relates to the previous term. If |$r$| > 1, the sequence grows; if |$r$| < 1, it decays.
• First Term ($a_1$): The starting point of the sequence, crucial for determining all subsequent terms.
• Convergence: For infinite geometric sequences, if |$r$| < 1, the sequence converges to a finite limit; otherwise, it diverges.

Examples of Geometric Sequences

Example 1: Find the 6th term of the geometric sequence where $a_1 = 5$ and $r = 3$.

Using the general term formula:

$$a_6 = 5 \times 3^{6-1} = 5 \times 243 = 1215$$

Example 2: Determine the general term of a geometric sequence where $a_1 = 2$ and $a_4 = 54$.

First, find the common ratio ($r$):

$$a_4 = a_1 \times r^{4-1} \Rightarrow 54 = 2 \times r^3 \Rightarrow r^3 = 27 \Rightarrow r = 3$$

Thus, the general term is:

$$a_n = 2 \times 3^{n-1}$$

Applications of Geometric Sequences

Geometric sequences model various real-world scenarios, including:

• Population Growth: Populations that grow by a constant percentage each period can be modeled using geometric sequences.
• Finance: Compound interest calculations often involve geometric sequences.
• Physics: Phenomena like radioactive decay follow geometric progression patterns.

Sum of Geometric Sequences

While not the primary focus, understanding the sum of geometric sequences complements the study of their general terms. The sum of the first $n$ terms ($S_n$) is given by:

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

This formula is particularly useful in applications like calculating the total amount accrued in finance over multiple periods.

Infinite Geometric Sequences

An infinite geometric sequence extends indefinitely. The sum to infinity ($S_\infty$) exists only if |$r$| < 1 and is calculated as:

$$S_\infty = \frac{a_1}{1 - r}$$

This concept is pivotal in fields like calculus and physics, where it aids in solving problems involving limits and converging series.

Common Mistakes to Avoid

• Confusing the common ratio with the difference in arithmetic sequences.
• Incorrectly applying the general term formula, especially misunderstanding the exponent.
• Neglecting to consider the absolute value of the common ratio when determining the behavior of the sequence.

Graphical Representation

Plotting a geometric sequence on a graph typically results in an exponential curve. If |$r$| > 1, the graph rises sharply, indicating rapid growth. If |$r$| < 1, the curve approaches the horizontal axis, showcasing decay.

Real-World Problem Solving

Understanding geometric sequences equips students with the skills to tackle complex problems. For instance, calculating the future value of an investment with compound interest relies on the principles of geometric progression.

Comparison Table

Summary and Key Takeaways