A sequence of terms in which each term is multiplied by a common ratio is called a geometric sequence.
Geometric Sequences Explained
A geometric sequence is an ordered list of numbers where the ratio between consecutive terms is constant. This constant ratio is referred to as the common ratio. According to the Lesson Summary, a geometric sequence is a sequence of numbers that is ordered with a specific pattern. Each successive number is the product of the previous number and a constant. The constant is the same for every term in the sequence and is called the common ratio.
Key Characteristics:
- Ordered: The terms appear in a specific order.
- Common Ratio: A fixed value is multiplied by each term to get the next term.
- Constant: The common ratio remains the same throughout the sequence.
Example:
Consider the sequence: 2, 6, 18, 54, ...
Here, each term is obtained by multiplying the previous term by 3. Therefore:
- 2 * 3 = 6
- 6 * 3 = 18
- 18 * 3 = 54
In this example, 3 is the common ratio, and the sequence is a geometric sequence.
Formula for the nth term:
The general formula for the nth term of a geometric sequence is:
an = a1 * r(n-1)
where:
- an is the nth term
- a1 is the first term
- r is the common ratio
- n is the term number
Practical Applications:
Geometric sequences appear in various real-world scenarios, including:
- Compound Interest: The amount of money grows geometrically if interest is compounded.
- Population Growth: Under ideal conditions, population growth can be modeled geometrically.
- Radioactive Decay: The amount of a radioactive substance decreases geometrically over time.
