Geometric Sequence Calculator
Find the nth term, the sum of n terms and the infinite sum of a geometric sequence from the first term and common ratio.
Last reviewed by the Radiatus Cloud team
Compute the nth term, sum of n terms and infinite sum of a geometric sequence a, ar, ar\u00b2, ...
Need this done properly for your business?
Radiatus delivers secure cloud, DevOps & compliance engineering.
Work with geometric sequences
A geometric sequence multiplies each term by a fixed number called the common ratio to get the next term. From the first term a with ratio r the sequence is a, a times r, a times r squared, and so on. This calculator finds the nth term using a times r to the power n minus one, the sum of the first n terms using a times one minus r to the n, all divided by one minus r, and the infinite sum a divided by one minus r when the ratio lies strictly between minus one and one.
When the absolute value of the ratio is one or more, the terms grow without bound and the infinite sum diverges, which the tool reports clearly.
Where geometric sequences matter
Geometric growth describes compound interest, population models, radioactive decay, repeated halving in algorithms, and the bouncing of a ball losing a fixed fraction of its height. The infinite-sum formula underlies many results in calculus and finance, such as the present value of a perpetual stream of payments.
Enter any real first term and ratio, including fractions and negatives, to explore growth, decay and alternating sequences. All calculation happens locally in your browser.
Related tools
- Percentage Calculator — Calculate percentages, percentage change, increase and decrease, and reverse percentages, with the working shown for each result.
- Statistical Calculator — Calculate mean, median, mode, standard deviation, and more.
- Probability Calculator — Calculate probabilities for single and combined events, including conditional probability and Bayes' theorem.
- Scientific Calculator — Perform advanced mathematical operations including trig, log, and exponents.
Frequently Asked Questions
What is the common ratio?
It is the fixed number each term is multiplied by to produce the next. A ratio above one grows the sequence; a ratio between zero and one shrinks it.
When does the infinite sum exist?
Only when the absolute value of the ratio is less than one. Then the terms shrink toward zero and the sum converges to a divided by one minus r.
What happens if the ratio is exactly one?
Every term equals the first term, so the sum of n terms is simply a times n, which the calculator handles as a special case.
Can the ratio be negative?
Yes. A negative ratio produces an alternating sequence whose terms switch sign, and the formulas still apply.
Why does my sum diverge?
If the absolute ratio is one or greater, the terms do not shrink, so adding infinitely many of them grows without limit and no finite infinite sum exists.
Privacy & Security
Everything runs in your browser; nothing is uploaded.
How to Use
Enter the first term, common ratio and number of terms to compute the sequence results.
Disclaimer: This tool is provided "as is" without warranty of any kind. Results are for educational and utility purposes.
Related Tools
Percentage Calculator
MathCalculate percentages, percentage change, increase and decrease, and reverse percentages, with the working shown for each result.
Statistical Calculator
MathCalculate mean, median, mode, standard deviation, and more.
Probability Calculator
MathCalculate probabilities for single and combined events, including conditional probability and Bayes' theorem.