Quick answer
In a geometric sequence aₙ = a₁rⁿ⁻¹ and the sum is Sₙ = a₁(1 − rⁿ) ÷ (1 − r). With a₁ = 3 and r = 2, the 5th term is 48 and the sum of five terms is 93.
Enter the first term, common ratio and term number. The calculator returns the nth term and the sum.
Related tools: Number Sequence Calculator: Identify an arithmetic or geometric sequence and find the next terms. Arithmetic Sequence Calculator: nth term and sum of an arithmetic sequence. Slope Calculator: Slope, angle, distance and line equation through two points.
How to use the Geometric Sequence Calculator
- Enter first term (a₁).
- Enter common ratio (r).
- Enter term number (n) (a whole number, between 1 and 1000).
- Press Calculate to see the result, and use Reset to start again.
| Input | What to enter | Allowed values |
|---|---|---|
| First term (a₁) | A number | — |
| Common ratio (r) | A number | — |
| Term number (n) | A whole number | ≥ 1, ≤ 1000 |
Formula and method
Formula
aₙ = a₁ rⁿ⁻¹; Sₙ = a₁(1 − rⁿ) ÷ (1 − r), r ≠ 1
IEEE-754 double precision; shown to at most 6 decimal places.
Closed-form formulas; r = 1 gives Sₙ = n a₁.
Worked examples
These examples are verified: each one is an automated test case that the live calculator must reproduce exactly.
Example 1
Inputs: First term (a₁) = 3; Common ratio (r) = 2; Term number (n) = 5.
| Result | Value |
|---|---|
| a5 | 48 |
| Sum of n terms | 93 |
Example 2
Inputs: First term (a₁) = 4; Common ratio (r) = 1; Term number (n) = 6.
| Result | Value |
|---|---|
| a6 | 4 |
| Sum of n terms | 24 |
Example 3
Inputs: First term (a₁) = 81; Common ratio (r) = 0.5; Term number (n) = 3.
| Result | Value |
|---|---|
| a3 | 20.25 |
| Sum of n terms | 141.75 |
The first line is the headline result (a5). Below it the geometric Sequence Calculator lists sum of n terms, which describe the same case from other angles. IEEE-754 double precision; shown to at most 6 decimal places.
Terms
3, 6, 12, 24, 48: each is twice the one before. S₅ = 3 × (1 − 32) ÷ (1 − 2) = 93, which matches 3 + 6 + 12 + 24 + 48.
Convergence
If |r| < 1 the infinite sum converges to a₁ ÷ (1 − r). For r = 1/2 and a₁ = 3 that is 6. Large ratios overflow and are rejected.
How the result changes with first term
Holding every other input at the first worked example, this table changes only first term. It shows how sensitive the answer is to that one value; each row was produced by running this calculator.
| First term | a5 | Sum of n terms |
|---|---|---|
| 1.5 | 24 | 46.5 |
| 3 | 48 | 93 |
| 6 | 96 | 186 |
| 12 | 192 | 372 |
What happens with invalid input
These messages are produced by the calculator itself, so you know what to fix.
- Common ratio (r): entering "1e200" shows "The result is too large." beside that field and no result.
Choosing between the Geometric Sequence Calculator and similar calculators
Each row shows what a related calculator on this site asks for and what it returns, so you can pick the one that matches your question.
| Calculator | You enter | It returns |
|---|---|---|
| [Number Sequence Calculator](/math/number-sequence-calculator/) | sequence | Pattern |
| [Arithmetic Sequence Calculator](/math/arithmetic-sequence-calculator/) | first term, common difference and term number | a10 |
| [Slope Calculator](/math/slope-calculator/) | x₁, y₁ and x₂ | Slope |
| [LCM Calculator](/math/lcm-calculator/) | numbers | Least common multiple |
| [Factor Calculator](/math/factor-calculator/) | number | Factors |
Background: algebra
Many algebra calculators rest on the same ideas: write the problem in a standard form, identify the coefficients, apply the formula and check the answer by substituting it back. For a quadratic ax² + bx + c = 0 the roots are (−b ± √(b² − 4ac)) ÷ 2a, and the discriminant b² − 4ac decides the kind of roots: positive gives two real roots, zero gives one repeated root and negative gives two complex conjugate roots.
If a = 0 the equation is linear, not quadratic. A sign slip on the −b term shifts both roots by the same amount, so substitute your roots back to check.
Assumptions and limitations
- Overflow rejected
If a required value is empty, not a number, or outside its allowed range, the form marks that field with a message and shows no result instead of a misleading number. The automated checks deliberately try invalid entries for common ratio (r) to confirm this.
Frequently asked questions
What is the common ratio?
The factor between successive terms.
What is the infinite sum?
a₁ ÷ (1 − r) for |r| < 1.
How does it differ from arithmetic?
Arithmetic adds, geometric multiplies.
What does the Geometric Sequence Calculator calculate?
nth term and sum of a geometric sequence. It returns a5 and Sum of n terms.
What formula does the Geometric Sequence Calculator use?
aₙ = a₁ rⁿ⁻¹; Sₙ = a₁(1 − rⁿ) ÷ (1 − r), r ≠ 1. Closed-form formulas; r = 1 gives Sₙ = n a₁.
What do I need to enter in the geometric Sequence Calculator?
You enter First term (a₁), Common ratio (r) and Term number (n).
How is the result rounded or limited?
IEEE-754 double precision; shown to at most 6 decimal places. Overflow rejected.
What is an example of a geometric Sequence result?
With first term (a₁) 3, common ratio (r) 2 and term number (n) 5, the a5 is 48.
Related calculators
- Number Sequence Calculator — Identify an arithmetic or geometric sequence and find the next terms.
- Arithmetic Sequence Calculator — nth term and sum of an arithmetic sequence.
- Slope Calculator — Slope, angle, distance and line equation through two points.
- LCM Calculator — Least common multiple of two or more numbers.
- Factor Calculator — List every factor of a positive whole number.
- Quadratic Formula Calculator — Solve ax² + bx + c = 0, including complex roots.
Sources
- OpenStax, Algebra and Trigonometry 2e (Rice University): exponents, radicals, logarithms and inverse trigonometric functions
Content last reviewed 2026-11-04. Formulas and worked examples are checked by automated tests against independently computed values.