A geometric sequence is a series of numbers where each term after the first is found by multiplying the previous term by a constant value, called the common ratio. Based on the provided reference, the sequence 2, 6, 18, 54, ... is a perfect example of a geometric sequence. Let's break down why.
Understanding Geometric Sequences
A geometric sequence, also called a geometric progression, is defined by its consistent ratio between consecutive terms. The general form for a geometric sequence is a, ar, ar², ar³, ..., where:
- 'a' is the first term,
- 'r' is the common ratio.
Example Breakdown: 2, 6, 18, 54, ...
Let's analyze the example from the reference: 2, 6, 18, 54, ...
- The first term ('a') is 2.
- To find the common ratio ('r'), divide any term by its preceding term.
- 6 / 2 = 3
- 18 / 6 = 3
- 54 / 18 = 3
- The common ratio ('r') is consistently 3.
This confirms that 2, 6, 18, 54, ... is indeed a geometric sequence because each term is obtained by multiplying the previous term by the fixed common ratio of 3.
Common Geometric Sequences in Various Fields
Geometric sequences pop up in many real-world scenarios including:
- Compound interest calculations: The amount of money grows geometrically over time if interest is compounded periodically.
- Population growth: Under ideal conditions, populations can grow geometrically.
- Radioactive decay: The amount of radioactive material decreases geometrically over time.
Key Features of a Geometric Sequence
- Constant Ratio: The ratio between any two consecutive terms is always the same.
- Multiplication: Each term is found by multiplying the previous term by the common ratio.
- Non-zero terms: Terms in geometric sequences are generally non-zero.
| Term | Value |
|---|---|
| 1st | 2 |
| 2nd | 6 |
| 3rd | 18 |
| 4th | 54 |
In summary, a clear example of a geometric sequence is 2, 6, 18, 54, ... as the ratio between each consecutive term is consistently 3.
