A geometric sequence is a list of numbers with a constant ratio between successive terms, while a geometric series is the sum of the terms in a geometric sequence.
Here's a more detailed breakdown:
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Geometric Sequence: This is an ordered list of numbers where each term is found by multiplying the previous term by a constant value called the common ratio (r).
- Example: 2, 4, 8, 16, 32... (Here, r = 2)
- General Term: an = a1 rn-1, where a1 is the first term, r is the common ratio, and n* is the term number.
-
Geometric Series: This is the sum of the terms in a geometric sequence.
- Example: 2 + 4 + 8 + 16 + 32... (This is the series formed from the sequence above)
- Finite Geometric Series Formula: Sn = a1 (1 - rn) / (1 - r), where Sn is the sum of the first n terms, a1 is the first term, r is the common ratio, and n* is the number of terms.
- Infinite Geometric Series Formula: S = a1 / (1 - r), where |r| < 1. This formula only works if the absolute value of the common ratio is less than 1, ensuring the series converges.
Here's a table summarizing the differences:
| Feature | Geometric Sequence | Geometric Series |
|---|---|---|
| Definition | A list of numbers with a constant ratio. | The sum of the terms in a geometric sequence. |
| Operation | Multiplication by a common ratio to get next term. | Addition of the terms in the sequence. |
| Representation | a1, a2, a3, ... | a1 + a2 + a3 + ... |
| Result | A list of numbers. | A single value (sum). |
In essence, a sequence is a collection of ordered numbers, whereas a series is the sum of those numbers. Think of a sequence as the ingredients and a series as the final dish made from those ingredients (where the "dish" is the sum).
