A pattern with a common ratio is a geometric sequence, where each term is found by multiplying the previous term by a constant value. This constant value is called the common ratio.
Understanding Geometric Sequences
A geometric sequence is characterized by a constant ratio between consecutive terms. This ratio, commonly denoted as r, determines how the sequence progresses.
- Definition: A sequence where the ratio between any two consecutive terms (an / an-1) is constant.
- Common Ratio (r): The constant value by which each term is multiplied to get the next term.
Example of a Geometric Sequence
Consider the sequence: 2, 6, 18, 54, ...
To find the common ratio (r), divide any term by its preceding term:
- r = 6 / 2 = 3
- r = 18 / 6 = 3
- r = 54 / 18 = 3
In this example, the common ratio is 3. Each term is obtained by multiplying the previous term by 3.
Formula for Geometric Sequences
The general formula for the nth term (an) of a geometric sequence is:
an = a1 * r(n-1)
where:
- a1 is the first term of the sequence
- r is the common ratio
- n is the term number
Key Characteristics
- Constant Multiplication: Each term is a constant multiple of the next term.
- Exponential Growth or Decay: If the common ratio is greater than 1, the sequence exhibits exponential growth. If the common ratio is between 0 and 1, the sequence exhibits exponential decay. If the common ratio is negative, the terms alternate in sign.
Real-World Applications
Geometric sequences appear in various real-world scenarios, including:
- Compound Interest: The growth of money in a savings account with compound interest follows a geometric sequence.
- Population Growth: Under ideal conditions, population growth can be modeled using a geometric sequence.
- Radioactive Decay: The decay of radioactive substances follows an exponential pattern, which is related to geometric sequences.
