Geometric Sequence Calculator
Calculate the nth term and finite sum of a geometric sequence from its first term, common ratio, and positive whole-number term count.
- Sum of n Terms
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Formula: aₙ = a₁rⁿ⁻¹; Sₙ = a₁(1 − rⁿ) ÷ (1 − r); when r = 1, Sₙ = na₁
Use a positive whole number for n. The first term and common ratio may be positive, negative or zero; results are rounded to four decimal places.
The Geometric Sequence Calculator finds the nth term and the finite sum of the first n terms from three inputs: the first term, common ratio and term count. Enter a finite first term, a finite common ratio and a positive whole number for n. The first term and ratio may be positive, negative or zero.
The default values are first term 2, common ratio 3 and n = 5. They form 2, 6, 18, 54, 162. The calculator returns an nth term of 162.0000 and a sum of 242.0000. These are mathematical sequence results, not forecasts of investment returns, loan costs or population change.
How to Use the Geometric Sequence Calculator
- Enter the first term of the sequence.
- Enter the fixed common ratio between consecutive terms.
- Enter n as a positive whole number.
- Select Calculate to view the nth term and finite sum.
Changing an input clears the earlier result until you calculate again. A negative ratio makes signs alternate. A zero ratio is also supported: after the first term, later terms are zero. Decimal first terms and ratios are accepted, while a decimal n is rejected because the sequence is indexed by positive integers.
Geometric Sequence Formulas and Worked Examples
aₙ = a₁ × r^(n − 1)
Sₙ = a₁ × (1 − r^n) ÷ (1 − r), r ≠ 1
Sₙ = n × a₁, r = 1Here, a₁ is the first term, r is the common ratio and n is the positive-integer term number. OpenStax’s College Algebra 2e section on geometric sequences defines the common ratio and gives the explicit nth-term formula. Its section on series and their notation derives the finite geometric-series formula for r ≠ 1.
For a negative-ratio example, first term 3, ratio −2 and n = 4 produce 3, −6, 12, −24. The nth term is −24.0000 and the sum is −15.0000. When r = 1, division by 1 − r is not used: every term equals the first term, so first term 4 and n = 6 return 4.0000 and 24.0000.
Input Rules, the Ratio-One Case and Related Calculators
Blank, nonnumeric and non-finite entries are invalid. n must be at least 1 and contain no fractional part. The ratio-one branch uses Sₙ = n × a₁, removing the closed-form formula’s division by zero while preserving the valid sequence.
Both outputs are displayed to four decimal places. Calculations use standard browser floating-point numbers, so long decimals may be rounded. Very large finite inputs can overflow to a non-finite intermediate or result; the widget then clears the results and displays an accessible calculation error. Display rounding does not add precision to the inputs.
Use the Arithmetic Sequence Calculator when terms change by a fixed difference. The Sum Calculator adds entered values, and the Proportion Calculator solves a fixed ratio equation. For related exponential arithmetic, see the Exponential Calculator or Exponential Growth Calculator; their assumptions and outputs differ from this sequence tool.
Frequently asked questions
What is a geometric sequence?
A geometric sequence is a sequence in which each term after the first is the previous term multiplied by one fixed common ratio.
What formulas does the calculator use?
It uses aₙ = a₁ × r^(n − 1) for the nth term. For the finite sum it uses Sₙ = a₁ × (1 − r^n) ÷ (1 − r) when r ≠ 1, and Sₙ = n × a₁ when r = 1.
Why must n be a positive whole number?
n identifies a position and a count of terms, so it must be an integer of at least 1. Zero, negative and fractional values of n are rejected.
Can the common ratio be zero or negative?
Yes. A zero ratio gives the first term followed by zeros, while a negative ratio alternates the signs of successive nonzero terms.
What happens when the common ratio is 1?
Every term equals the first term. The nth term is a₁ and the finite sum is n × a₁, so the calculator uses that direct branch instead of dividing by zero.
How are results rounded, and what happens with very large inputs?
Results are displayed to four decimal places using browser floating-point arithmetic. If the calculation becomes non-finite, the widget clears the results and displays a calculation error.
Does this calculator predict finance or population outcomes?
No. It evaluates a mathematical geometric sequence from the values entered. Real forecasts require assumptions and models that this calculator does not provide.