A geometric sequence differs from an arithmetic sequence in the fundamental operation used to generate subsequent terms. While arithmetic sequences rely on addition or subtraction of a common difference, geometric sequences use multiplication or division by a common ratio, as stated in the reference.
Key Differences Summarized
Here's a table highlighting the key differences:
| Feature | Arithmetic Sequence | Geometric Sequence |
|---|---|---|
| Definition | Constant difference between consecutive terms | Constant ratio between consecutive terms |
| Operation | Addition or Subtraction (of common difference) | Multiplication or Division (by common ratio) |
| Formula (General) | an = a1 + (n-1)d | an = a1 * r(n-1) |
| 'd' or 'r' | Common difference (d) | Common ratio (r) |
Elaboration on the Differences
Arithmetic Sequences
- Definition: An arithmetic sequence is a list of numbers where the difference between any two consecutive terms is constant. This constant difference is called the common difference.
- Example: 2, 5, 8, 11, 14... (Common difference = 3) To get the next term, you add 3 to the previous term.
Geometric Sequences
- Definition: A geometric sequence is a list of numbers where the ratio between any two consecutive terms is constant. This constant ratio is called the common ratio. According to the reference, "Geometric sequences are defined by an initial value and a common ratio, with the same number multiplied or divided to each term".
- Example: 3, 6, 12, 24, 48... (Common ratio = 2) To get the next term, you multiply the previous term by 2. Another example could be 100, 50, 25, 12.5,... (Common ratio = 0.5 or 1/2), showcasing division.
Practical Insights
- Growth Rate: Geometric sequences generally exhibit much faster growth or decay than arithmetic sequences, especially when the common ratio is significantly greater than 1 or significantly less than -1.
- Applications: Arithmetic sequences are often used to model linear growth or decay scenarios, while geometric sequences are used to model exponential growth or decay scenarios like compound interest, population growth, or radioactive decay.
