Geometric Sequence Calculator
Calculate the nth term, partial sum, and infinite sum of a geometric sequence defined by its first term and common ratio.
Each term is multiplied by this value
Minimum: 1. Whole numbers only
Results
nth Term (aₙ)
4374
Partial Sum (Sₙ)
6560
Infinite Sum (S∞)
∞ (diverges; |r| ≥ 1)
How to Use This Calculator
Enter your input values
Fill in all required input fields for the Geometric Sequence Calculator. Follow the units and input format printed next to each field. Where a unit selector is available, choose the units that match your data.
Review your inputs
Double-check that all values are correct and match the units or format shown for each field. Incorrect units are the most common source of calculation errors and can produce results that are off by factors of 2, 10, or more.
Read the results
The Geometric Sequence Calculator computes the output in your browser and displays the results, with units clearly labeled, as soon as they are ready — no data sent to a server.
Explore parameter sensitivity
Try adjusting individual input values to see how the output changes. This is a quick and effective way to develop intuition about how different parameters influence the result and to identify which inputs have the largest effect.
When to Use This Calculator
- •Use the Geometric Sequence Calculator when you need a quick mathematical result without writing out all the steps manually, saving time on repetitive calculations.
- •Use it to verify hand calculations on tests or assignments and catch arithmetic mistakes.
- •Use it when teaching or explaining mathematical concepts to others, demonstrating how changing inputs affects the result.
- •Use it to explore the behavior of mathematical functions across a range of inputs.
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About Geometric Sequence Calculator
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. Geometric sequences model exponential growth and decay, making them indispensable in finance (compound interest), biology (population growth), physics (radioactive decay), and computer science (algorithmic complexity). Unlike arithmetic sequences that grow linearly, geometric sequences grow (or shrink) exponentially. When the absolute value of the common ratio is less than one, the infinite series converges to a finite value, a result that underpins much of calculus and signal processing. This calculator computes the nth term, the sum of the first n terms, and — when applicable — the sum of the infinite series, giving you a complete picture of any geometric sequence.
The Math Behind It
Formula Reference
nth Term
aₙ = a₁ × r^(n−1)
Variables: a₁ = first term, r = common ratio, n = term number
Partial Sum
Sₙ = a₁(1 − rⁿ) / (1 − r), r ≠ 1
Variables: n = number of terms
Infinite Sum
S∞ = a₁ / (1 − r), |r| < 1
Variables: Converges only when |r| < 1
Worked Examples
Example 1: Find the 6th Term
Given a₁ = 5 and r = 2, find a₆.
a₆ = 160
Example 2: Sum of an Infinite Geometric Series
Given a₁ = 10 and r = 0.5, find S∞.
S∞ = 20
Common Mistakes & Tips
- !Using the partial sum formula when r = 1 (divide-by-zero); use Sₙ = n × a₁ instead.
- !Attempting to calculate S∞ when |r| ≥ 1 — the series diverges.
- !Confusing the exponent: the nth term uses r^(n−1), not r^n.
- !Forgetting that a negative ratio produces alternating signs.
Related Concepts
Used in These Calculators
Calculators that build on or apply the concepts from this page:
Frequently Asked Questions
When does an infinite geometric series converge?
An infinite geometric series converges if and only if the absolute value of the common ratio is less than 1 (|r| < 1). When it converges, the sum is S∞ = a₁/(1 − r).
Can the common ratio be negative?
Yes. A negative common ratio means the terms alternate between positive and negative values. For example, with a₁ = 1 and r = −2, the sequence is 1, −2, 4, −8, 16, ….
How is compound interest related to geometric sequences?
Each compounding period multiplies the balance by (1 + rate), producing a geometric sequence of balances. After n periods: Balance = Principal × (1 + rate)ⁿ.
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