# Geometric Sequence vs. Exponential Function — What's the Difference?

By Tayyaba Rehman — Published on January 9, 2024
A geometric sequence is a series of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number. An exponential function is a mathematical expression where a constant base is raised to a variable exponent.

## Key Differences

A geometric sequence is a set of numbers where each term is produced by multiplying the previous term by a constant ratio. This ratio is called the common ratio and remains the same throughout the sequence. For example, in the sequence 2, 4, 8, 16, each term is twice its predecessor.
An exponential function is a more general mathematical concept where a constant base is raised to the power of a variable exponent. This exponent dictates how the function grows or decays, and it is not limited to whole numbers. Exponential functions model growth or decay processes, like population growth or radioactive decay.
In a geometric sequence, the pattern is discrete, meaning it is defined at specific points, typically at integer intervals. Each term of the sequence can be represented by a formula involving the initial term and the common ratio to the power of the position in the sequence minus one.
Exponential functions, however, provide a continuous model. They are defined for all real numbers and are represented by a continuous curve on a graph. The base of an exponential function does not change, similar to the common ratio in a geometric sequence, but its power does, allowing for a continuous range of values.
While both geometric sequences and exponential functions deal with repeated multiplication, a geometric sequence is a function on the integers (discrete), and an exponential function is defined for all real numbers (continuous). Moreover, geometric sequences are often used to model situations where discrete jumps or steps occur, while exponential functions fit scenarios where changes occur continuously over time.

## Comparison Chart

### Definition

A sequence with a constant ratio between successive terms.
A function with a constant base raised to a variable exponent.

### Discreteness/Continuity

Discrete; defined at specific points, often integers.
Continuous; defined for all real numbers.

### Representation

As a list or set of numbers.
As a mathematical expression or graph.

### Application

Often in scenarios with discrete growth or decay.
Typically models continuous growth or decay processes.

### Example

The amounts of a drug in the bloodstream at hourly intervals.
The continuous growth of bacteria over time.

## Compare with Definitions

#### Geometric Sequence

An ordered list where each term after the first is found by multiplying the previous term by a constant.
The sequence 5, 15, 45, ... is geometric with a common ratio of 3.

#### Exponential Function

An equation that represents continuous growth or decay.
The amount of a radioactive substance decreases according to the exponential function y = 100

#### Geometric Sequence

A pattern of numbers that follow a multiplicative path.
In her savings plan, each month's saving is twice that of the previous month, forming a geometric sequence.

#### Exponential Function

The exponential function y = e^x is often used to model natural growth processes.
A representation of a series where the exponent is a variable.

#### Geometric Sequence

A series of numbers multiplied by a common ratio.
If the first term of a geometric sequence is 1 and the common ratio is 3, the sequence is 1, 3, 9, 27, ...

#### Exponential Function

A mathematical function where a number is raised to a variable power.
The formula y = 2^x describes an exponential function.

#### Geometric Sequence

A progression where each term is the previous term times a fixed number.
The sequence of 2, 6, 18, ... has each term three times larger than the one before.

#### Exponential Function

0.5^t.
A function where the rate of change increases or decreases exponentially.

#### Geometric Sequence

Successive terms increase by a fixed proportion.
The geometric sequence representing the population of a species is 100, 300, 900, ...

#### Exponential Function

For the exponential function f
X = 3^x, f

#### Exponential Function

The exponential function representing the investment return is y = P
1+r^t.

#### Exponential Function

2 is 9.
A function with a constant base raised to the power of the independent variable.

## Common Curiosities

#### How is an exponential function defined?

An exponential function is defined as a function with a constant base raised to the power of a variable exponent.

#### What are exponential functions used for?

Exponential functions are used to model a variety of real-world phenomena that exhibit exponential growth or decay, such as population growth, finance, and physics.

#### Can a geometric sequence be decreasing?

Yes, if the common ratio between terms is between 0 and 1, the geometric sequence will decrease.

#### How does the graph of an exponential function appear?

The graph of an exponential function is a smooth curve that continuously increases or decreases, depending on the base.

#### What happens when the base of an exponential function is greater than 1?

When the base is greater than 1, the exponential function will show exponential growth.

#### Do all geometric sequences converge?

No, geometric sequences converge only if the absolute value of the common ratio is less than one.

#### Is a geometric sequence a type of exponential function?

While similar in concept, a geometric sequence is discrete and typically applies to integers, while an exponential function is continuous and applies to all real numbers.

#### What is a geometric sequence?

A geometric sequence is a series where each term is found by multiplying the previous term by a fixed number called the common ratio.

#### Can a geometric sequence have a negative common ratio?

Yes, a geometric sequence can have a negative common ratio, which causes the terms to alternate in sign.

#### What is the common ratio in a geometric sequence?

The common ratio is the factor by which consecutive terms of the sequence are multiplied.

#### Are exponential functions always increasing?

Not always; if the base is between 0 and 1, the exponential function is decreasing.

#### What does the graph of a geometric sequence look like?

The graph of a geometric sequence consists of distinct points at regular intervals along the x-axis.

#### Can exponential functions only model growth?

No, exponential functions can model both growth and decay, depending on the base and the exponent.