An example of a geometric sequence is 2, 4, 8, 16, .... This is a geometric sequence because the ratio between consecutive terms is constant (it's always 2).
Understanding Geometric Sequences
A geometric sequence is a list 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.
- Key Features:
- Each term is the product of the previous term and the common ratio.
- The common ratio remains constant throughout the sequence.
- Examples from various sources:
- 5, 10, 20, 40, 80,... (common ratio = 2)
- 2, 10, 50, 250,... (common ratio = 5)
- 10, 5, 2.5, 1.25,... (common ratio = 0.5)
- 1, 3, 9, 27,... (common ratio = 3)
How to Identify a Geometric Sequence
To determine if a sequence is geometric, calculate the ratio between consecutive terms. If the ratio is consistent, it's a geometric sequence.
For example, in the sequence 2, 6, 18, 54,...:
- 6/2 = 3
- 18/6 = 3
- 54/18 = 3
The common ratio is 3; therefore, this is a geometric sequence.
Applications of Geometric Sequences
Geometric sequences appear in various mathematical contexts and real-world scenarios, including:
- Compound interest: The growth of money invested with compound interest follows a geometric sequence.
- Population growth (under ideal conditions): If a population increases by a fixed percentage each year, its growth can be modeled using a geometric sequence.
- Radioactive decay: The decay of radioactive substances is often modeled using a geometric sequence (with a common ratio less than 1).
