An example of a common ratio in a geometric sequence is 2.
A geometric sequence is characterized by a constant ratio between consecutive terms. This constant ratio is known as the common ratio. In other words, to get the next term in a geometric sequence, you multiply the current term by the common ratio.
Understanding Common Ratios
The common ratio can be any non-zero number, positive or negative, integer or fraction. Let's explore how it works:
- Positive Common Ratio: A sequence like 2, 4, 8, 16, 32, ... has a common ratio of 2. Each term is obtained by multiplying the previous term by 2 (2 2 = 4, 4 2 = 8, and so on).
- Negative Common Ratio: A sequence like 1, -3, 9, -27, 81, ... has a common ratio of -3.
- Fractional Common Ratio: A sequence like 1, 1/2, 1/4, 1/8, ... has a common ratio of 1/2.
- Non-integer Common Ratio: A sequence like 1, 1.5, 2.25, 3.375,... has a common ratio of 1.5.
Calculating the Common Ratio
To find the common ratio, you can simply divide any term in the sequence by the term that precedes it. For example:
| Sequence | Terms | Common Ratio Calculation | Common Ratio |
|---|---|---|---|
| 2, 4, 8, 16, 32, ... | 4 and 2 | 4 / 2 = 2 | 2 |
| 1, -3, 9, -27, ... | -3 and 1 | -3 / 1 = -3 | -3 |
| 1, 1/2, 1/4, 1/8,... | 1/2 and 1 | (1/2) / 1 = 1/2 | 1/2 |
| 1, 1.5, 2.25, 3.375,... | 1.5 and 1 | 1.5 / 1 = 1.5 | 1.5 |
Importance of the Common Ratio
The common ratio plays a crucial role in defining the characteristics of a geometric sequence. It dictates whether the terms of the sequence increase or decrease, and whether they alternate in sign.
Based on the provided reference, in the sequence 2, 4, 8, 16, 32, each term is obtained by multiplying the previous term by 2. Therefore, 2 is a common ratio of a geometric sequence.
