If you need the nth term, the common ratio, or the sum of terms in a geometric sequence, this Geometric Sequence Calculator computes the result instantly. Enter the values you know, choose what you want to find, and get a correct answer with units handled consistently.
A geometric sequence changes by multiplying each term by the same number. Once you know the first term and the common ratio, every other term and sum follows from one clean set of formulas.
What Is a Geometric Sequence?
A geometric sequence is a list of numbers where the ratio between consecutive terms is constant. That constant is the common ratio, usually written as r.
- First term: a (often written as a₁)
- Common ratio: r
- nth term: aₙ
Example: If a₁ = 3 and r = 2, the sequence is 3, 6, 12, 24, …. Each term is the previous term multiplied by 2.
Core Formulas (The Heart of the Calculator)
Geometric sequence formulas connect the inputs (like a, r, and n) to outputs (like aₙ and sums). The calculator uses these exact relationships.
1) nth Term Formula
The nth term of a geometric sequence is:
aₙ = a · r^(n−1)
- a is the first term (a₁)
- r is the common ratio
- n is the term number (n ≥ 1)
2) Common Ratio From Two Terms
If you know two terms, you can solve for the common ratio. For terms a₁ and aₖ:
r = (aₖ / a₁)^(1/(k−1))
Special case: if a₁ = 0, then the ratio depends on whether later terms are also 0. The calculator handles invalid/undefined cases with clear messages.
3) Sum of a Finite Geometric Series
The sum of the first n terms is:
Sₙ = a · (1 − r^n) / (1 − r) for r ≠ 1
If r = 1, every term equals a, so:
Sₙ = a · n
- Sₙ is the total of terms 1 through n
- r controls whether the series grows (|r| > 1) or shrinks (|r| < 1)
How to Use the Geometric Sequence Calculator
Use the calculator to compute one target at a time. The calculator’s “mode” selector determines which formulas it applies and which inputs are required.
- Choose what to find: nth term, common ratio, or sum.
- Enter known values: first term, common ratio (if needed), term number, and/or another term.
- Click Calculate: the result appears in a formatted results box.
Invalid inputs (like non-numeric values, n less than 1, or undefined ratio situations) trigger an error message and highlight the field in red.
Worked Example 1: Finding the nth Term
Suppose a sequence starts with a₁ = 5 and multiplies by r = 3 each step. Find the 10th term.
- a = 5
- r = 3
- n = 10
Using aₙ = a · r^(n−1):
a₁₀ = 5 · 3^9
That value is large, but the calculator computes it accurately.
Worked Example 2: Finding the Sum of First Terms
Imagine a value grows by 10% each step, so r = 1.1. If the first term is a₁ = 200, what is the sum of the first 6 terms?
- a = 200
- r = 1.1
- n = 6
Because r ≠ 1, use:
Sₙ = a · (1 − r^n) / (1 − r)
This is common in finance and growth problems where you want the total across multiple steps.
Practical Use-Cases (Where Geometric Sequences Show Up)
1) Finance: Compound Growth
Many real-world growth processes compound by a fixed rate. If an account grows by a constant percentage each period, the balances form a geometric sequence.
- Balance after n periods: use the nth term
- Total accumulated over n periods: use the sum
2) Science and Engineering: Decay and Repeats
Geometric sequences model repeated multiplication, including decay (like radioactive materials approximations) and signal attenuation (like repeated loss through components).
- Decay over time: common ratio less than 1
- Total effect after n steps: the finite sum
Frequently Asked Questions
What is the difference between a geometric sequence and an arithmetic sequence?
A geometric sequence multiplies by a constant ratio r to move from one term to the next. An arithmetic sequence adds a constant difference d instead. Because one uses multiplication and the other uses addition, their formulas for the nth term and sums are different.
How do I find the common ratio if I know two terms?
Use the ratio formula r = (aₖ / a₁)^(1/(k−1)). Here a₁ is the first term and aₖ is the kth term. This works when a₁ is not zero. If a₁ is zero, the ratio may be undefined.
When is the sum formula for a geometric series equal to a·n?
The sum of the first n terms becomes Sₙ = a·n when the common ratio r equals 1. In that case, every term is the same value a, so you simply add a to itself n times. For r ≠ 1, use the fraction formula.
Can geometric sequences have negative terms?
Yes. A negative common ratio r produces alternating signs, creating sequences like 4, −8, 16, −32, … The nth term formula still works, and the sum formula works for finite n. Just be careful with powers and signs.
Why does the calculator sometimes show an error?
Errors appear when the requested calculation is mathematically undefined or inputs are invalid. Examples include n less than 1, non-numeric entries, or trying to compute a ratio that requires dividing by zero. Fix the highlighted field and try again for a valid result.
Quick Reference Table
| Goal | Formula | Inputs You Need |
|---|---|---|
| nth term | aₙ = a · r^(n−1) | a, r, n |
| common ratio | r = (aₖ / a₁)^(1/(k−1)) | a₁, aₖ, k |
| sum of first n terms | Sₙ = a·(1 − r^n)/(1 − r) (r ≠ 1), else Sₙ = a·n | a, r, n |
Use the calculator above for fast, accurate results. If you want to check your work by hand, the formulas in this article match what the calculator computes.
