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Geometric Sequence Calculator
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Date publish: 16.09.2024
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Author: Calcwizard
Understanding Geometric Sequences
A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. For instance, if the first term is 5 and the common ratio is 3, the sequence would be 5, 15, 45, 135, …
Key Features of Geometric Sequences
- The ratio between consecutive terms is constant.
- They can model exponential growth or decay in real-world scenarios.
- The nth term can be calculated using the formula: an = a1 * r(n-1), where a1 is the first term, r is the common ratio, and n is the term number.
Examples of Geometric Sequences
| a1 (First Term) | r (Common Ratio) | n (Term Number) | an (Nth Term) |
|---|---|---|---|
| 2 | 2.0 | 5 | 32.000 |
| 3 | 0.5 | 4 | 0.375 |
| 1 | 3.0 | 6 | 243.000 |
| 5 | 4.0 | 3 | 80.000 |
| 4 | 1.5 | 5 | 20.250 |
| 6 | 0.8 | 7 | 1.31072 |
| 10 | 2.5 | 4 | 156.250 |
| 7 | 1.2 | 6 | 17.30496 |
Interesting Facts
- Geometric sequences are used in finance to calculate compound interest.
- They appear in nature, such as in the branching of trees or the arrangement of leaves on a stem.
- The sum of a geometric series can be calculated using the formula: Sn = a1 * (1 – rn) / (1 – r) for |r| < 1.
