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How to use
A geometric sequence multiplies each term by a common ratio r. The calculator returns the nth term, finite sum and first terms, and only shows an infinite sum when |r|<1 because that is the convergence condition.
- Enter first term a₁.
- Enter common ratio r; it may be negative or zero.
- Enter positive integer n.
- Review the nth term and finite sum, plus infinite sum when convergent.
How it is calculated
Example
With a₁=2, r=3 and n=5, the fifth term is 162 and the first-five-term sum is 242. If r=0.5 instead, the infinite sum is 4.
Important notes
The r=1 case is handled separately to avoid division by zero. A finite infinite-series sum is not shown for |r|≥1 under the standard geometric-series model.
Worked examples and interpreting results
A geometric sequence multiplies each term by the same ratio. Beginning at 3 with ratio 2 gives 3, 6, 12, and 24.
| Case | Calculation | Result |
|---|---|---|
| Fourth term | 3 × 2³ | 24 |
| Sum of four terms | 3 + 6 + 12 + 24 | 45 |
How to check the result
An infinite sum exists only if the absolute common ratio is below 1. If the ratio equals 1, the finite sum is a₁ × n.
Frequently asked questions
When does the infinite series converge?
When |r|<1.
What happens at r=1?
Every term equals a₁ and Sₙ=na₁.
Can r be negative?
Yes; the term signs may alternate.
What happens when the common ratio is negative or zero?
A negative ratio makes term signs alternate, while magnitudes follow the absolute ratio. If the ratio is zero, every term after the first is zero, and special care is needed when using sum formulas.