MihsabaHow 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
aₙ=a₁r^(n−1); Sₙ=a₁(1−rⁿ)÷(1−r), and for r=1: Sₙ=na₁
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.
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.
Is this calculator free?
Yes. It is free to use and requires no account.