You find geometric patterns, specifically geometric sequences, by identifying a common ratio that multiplies each term to get the next one.
Geometric patterns can appear in various forms, but one fundamental type, particularly in mathematics, is the geometric sequence. These are patterns found in series of numbers where the relationship between consecutive terms involves multiplication.
Understanding Geometric Sequences
A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.
As stated in the reference:
Geometric sequences use multiplication to find each subsequent term. Each term gets multiplied by a common ratio, resulting in the next term in the sequence.
Finding the Common Ratio
The key to finding a geometric sequence pattern is determining the common ratio. To do this, you can divide any term in the sequence by its preceding term.
- Method:
- Pick any term in the sequence (except the first).
- Divide that term by the term immediately before it.
- Repeat this for at least one other pair of consecutive terms. If the result is the same each time, that value is the common ratio.
If a common ratio exists consistently throughout the sequence, then you have identified a geometric sequence pattern.
Example
Consider the sequence: 3, 6, 12, 24, 48, ...
- Divide the second term by the first: $6 \div 3 = 2$
- Divide the third term by the second: $12 \div 6 = 2$
- Divide the fourth term by the third: $24 \div 12 = 2$
Since the result is consistently 2, the common ratio is 2. This confirms that the sequence is a geometric sequence, and the pattern is "multiply by 2" to find the next term. The reference provides a similar example, noting "In the geometric sequence shown below, the common ratio is 2. In other words, each term is multiplied by 2."
By identifying this common ratio through division, you uncover the multiplicative rule that defines the geometric sequence pattern.
