Sum of a Geometric Sequence

Introduction

Key Concepts

Definition of a Geometric Sequence

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 ($r$). Formally, a geometric sequence can be expressed as:

$$ a, \; ar, \; ar^2, \; ar^3, \; \dots $$

where:

• $a$ is the first term of the sequence.
• $r$ is the common ratio.

For example, the sequence 2, 6, 18, 54, ... is geometric with $a = 2$ and $r = 3$.

Formula for the Sum of the First $n$ Terms

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

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

If the common ratio $r$ is 1, the sum simplifies to:

$$ S_n = a \cdot n $$

Derivation of the Sum Formula

To derive the sum formula, consider the sum of the first $n$ terms:

$$ S_n = a + ar + ar^2 + ar^3 + \dots + ar^{n-1} $$

Multiplying both sides by $r$:

$$ rS_n = ar + ar^2 + ar^3 + \dots + ar^{n-1} + ar^n $$>

Subtract the second equation from the first:

$$ S_n - rS_n = a - ar^n $$>

Factor out $S_n$ and $a$:

$$ S_n(1 - r) = a(1 - r^n) $$>

Solving for $S_n$:

$$ S_n = a \cdot \frac{1 - r^n}{1 - r} $$>

Infinite Geometric Series

When the number of terms in a geometric sequence approaches infinity and the absolute value of the common ratio is less than one ($|r| < 1$), the series converges to a finite sum:

$$ S = \frac{a}{1 - r}, \quad \text{for } |r| < 1 $$>

This is particularly useful in applications where the sequence continues indefinitely but the sum remains bounded.

Applications of Geometric Series

Geometric series have numerous applications across different fields:

• Finance: Calculating compound interest involves geometric sequences.
• Population Growth: Modeling populations with consistent growth rates.
• Physics: Analyzing phenomena like radioactive decay.
• Computer Science: Assessing algorithm complexities and recursion.

Examples

Example 1: Find the sum of the first 5 terms of the geometric sequence where $a = 3$ and $r = 2$.

Solution:

$$ S_5 = 3 \cdot \frac{1 - 2^5}{1 - 2} = 3 \cdot \frac{1 - 32}{-1} = 3 \cdot 31 = 93 $$>

Example 2: Determine the sum to infinity of the geometric series with $a = 4$ and $r = \frac{1}{3}$.

Solution:

$$ S = \frac{4}{1 - \frac{1}{3}} = \frac{4}{\frac{2}{3}} = 6 $$>

Since $|r| < 1$, the infinite series converges to 6.

Properties of Geometric Series

• Common Ratio: Determines the growth or decay of the sequence.
• Convergence: Infinite series converges only if $|r| < 1$.
• Exponential Growth/Decay: Geometric sequences exhibit exponential behavior based on $r$.

Sum in Terms of Initial and Final Terms

Sometimes, the sum of a geometric sequence is expressed in terms of the initial term ($a$) and the final term ($l$):

$$ S_n = \frac{a(1 - r^n)}{1 - r} = \frac{a - l}{1 - r} $$>

where $l = ar^{n-1}$.

Relationship with Arithmetic Sequence

While geometric sequences involve multiplication by a common ratio, arithmetic sequences involve addition of a common difference. This fundamental difference leads to distinct behaviors and applications for each type of sequence.

Graphical Representation

The graph of a geometric sequence is exponential. If $r > 1$, the sequence grows exponentially; if $0 < r < 1$, it decays exponentially. Negative values of $r$ result in alternating sequences.

Common Misconceptions

• Confusing geometric sequences with arithmetic sequences.
• Incorrectly applying the sum formula without considering the value of $r$.
• Assuming all infinite geometric series converge.

Comparison Table

Aspect	Geometric Sequence	Arithmetic Sequence
Definition	Each term is multiplied by a constant ratio ($r$).	Each term is obtained by adding a constant difference ($d$).
General Form	$a, \; ar, \; ar^2, \; \dots$	$a, \; a + d, \; a + 2d, \; \dots$
Sum Formula	$S_n = a \cdot \frac{1 - r^n}{1 - r}$	$S_n = \frac{n}{2}[2a + (n - 1)d]$
Growth Pattern	Exponential growth or decay based on $r$.	Linear growth or decay based on $d$.
Applications	Finance, population models, physics.	Scheduling, simple interest calculations, distance problems.
Convergence of Infinite Series	Converges if $|r| < 1$.	Never converges; always diverges.

Summary and Key Takeaways