1. Definition of a Geometric Sequence
A geometric sequence is a list of numbers where each term is obtained by multiplying the previous term by a fixed number. This fixed number is called the common ratio.
Example:
\( 3,\ 9,\ 27,\ 81,\ldots \)
Each term is obtained by multiplying the previous term by 3, so the common ratio is \( r = 3 \).
2. Common Ratio
The common ratio tells how the sequence changes from one term to the next. It is the key feature that defines a geometric sequence.
2.1. Meaning of Common Ratio \(r\)
The common ratio \( r \) is the number multiplied by each term to get the next term.
It is given by:
\( r = \dfrac{a_{n}}{a_{n-1}} \)
If \( r > 1 \), the sequence grows. If \( 0 < r < 1 \), the sequence shrinks.
2.2. Finding \(r\) from Terms
You can find the common ratio by dividing a term by the term before it.
Example:
Sequence: \( 16,\ 8,\ 4,\ 2,\ldots \)
\( r = \dfrac{8}{16} = \dfrac{1}{2} \)
A ratio less than 1 means the sequence is decreasing.
3. General Term of a Geometric Sequence
The general term of a geometric sequence helps find any term directly without listing previous terms.
3.1. Formula for \(a_n\)
If \( a_1 \) is the first term and \( r \) is the common ratio, then the nth term is:
\( a_n = a_1 \cdot r^{n-1} \)
This formula helps jump to any term, even a very large one.
3.2. Using \(a_n\) to Find Unknown Terms
The formula can also be used to find missing information such as the common ratio or earlier terms.
Example:
Given \( a_5 = 162 \) and \( a_1 = 2 \):
\( 162 = 2 \cdot r^{4} \)
\( r^{4} = 81 \Rightarrow r = 3 \)
4. Behaviour of a Geometric Sequence
The behaviour of a geometric sequence depends entirely on the value of the common ratio \( r \). Different values of \( r \) create very different patterns.
4.1. Case \(|r| > 1\): Growth
When \( |r| > 1 \), the sequence grows rapidly because each term becomes larger than the previous one.
Example:
\( 2,\ 6,\ 18,\ 54,\ldots \)
Here, \( r = 3 \) and the sequence grows quickly.
4.2. Case \(|r| < 1\): Decay
When \( |r| < 1 \), each term becomes smaller. This is called geometric decay.
Example:
\( 80,\ 40,\ 20,\ 10,\ldots \)
Here, \( r = \dfrac{1}{2} \).
4.3. Negative Common Ratio
If the common ratio is negative, the sequence alternates between positive and negative terms.
Example:
\( 4,\ -8,\ 16,\ -32,\ldots \)
The signs flip each time because \( r = -2 \).
5. Examples of Geometric Sequences
Geometric sequences appear naturally in patterns involving repeated multiplication, doubling, or halving.
5.1. Simple GP Examples
- \( 5, 15, 45, 135, \ldots \)
- \( 1, \dfrac{1}{3}, \dfrac{1}{9}, \dfrac{1}{27}, \ldots \)
- \( -2, 6, -18, 54, -162, \ldots \)
These patterns highlight multiplication-based growth or oscillation.
5.2. Real-Life GP Patterns
Many real situations follow geometric sequences, such as:
- Population growth under steady multiplication
- Radioactive decay with constant reduction
- Interest growth in savings with fixed percentage rates
These show how geometric sequences model rapid growth or steady decline.