Calculate the nth term
aₙ = a₁ × qⁿ⁻¹Sequence formulas
Geometric sequence
Calculate the nth term, the sum of the first n terms, or the sum of a convergent infinite geometric series, and work backwards to a₁, q, or n.
Formula
aₙ = a₁ × qⁿ⁻¹; Sₙ = a₁ × (1 - qⁿ) / (1 - q) In a geometric sequence every term is the previous one multiplied by the same ratio q, so the nth term grows exponentially; the partial sum needs a separate formula for q = 1, and the whole infinite series has a finite sum only for |q| < 1.
Geometric sequence calculator
Symbols and units
| Symbol | Quantity | SI | Supported units |
|---|---|---|---|
| a₁ | first termValue of the term with index 1; it must be different from zero. | 1 | 1 |
| q | common ratioConstant quotient of two consecutive terms: q = aₙ₊₁ / aₙ. It must be different from zero and may be negative. | 1 | 1 |
| n | term indexPosition of the term in the sequence; a natural number greater than or equal to 1. | 1 | 1 |
| aₙ | nth termValue of the term with index n, calculated from the explicit formula. | 1 | 1 |
| Sₙ | sum of the first n termsSum a₁ + a₂ + ... + aₙ; for q = 1 it reduces to n × a₁. | 1 | 1 |
| S | sum of the infinite seriesLimit of the partial sums, defined only for |q| < 1. | 1 | 1 |
Formula rearrangements
Sum of n terms for q ≠ 1
Sₙ = a₁ × (1 - qⁿ) / (1 - q)Sum of n terms for q = 1
Sₙ = n × a₁Sum of the infinite series for |q| < 1
S = a₁ / (1 - q)Calculate the first term
a₁ = aₙ / qⁿ⁻¹Calculate the common ratio
q = (aₙ / a₁)^(1 / (n - 1))Worked examples
Eighth term of a sequence
A geometric sequence starts with a₁ = 3 and has a common ratio q = 2. Calculate a₈.
- 1Step 1: Use the explicit formula aₙ = a₁ × qⁿ⁻¹.
- 2Step 2: Substitute a₁ = 3, q = 2 and n = 8: a₈ = 3 × 2⁷.
- 3Step 3: Calculate 2⁷ = 128 and multiply by 3 to obtain a₈ = 384.
Result: a₈ = 384
Sum of a convergent infinite series
Calculate the sum of the infinite series 4 + 2 + 1 + 0.5 + ..., where a₁ = 4 and q = 0.5.
- 1Step 1: Check convergence: |q| = 0.5 < 1, so the series has a finite sum.
- 2Step 2: Use the formula S = a₁ / (1 - q).
- 3Step 3: Substitute: S = 4 / (1 - 0.5) = 4 / 0.5 = 8.
Result: S = 8
Sources and review
- Algebra and Trigonometry 2e, section 13.3: Geometric Sequences
OpenStax · Reviewed 29 August 2026
Definition of a geometric sequence and the explicit formula for its nth term.
- Algebra and Trigonometry 2e, section 13.4: Series and Their Notations
OpenStax · Reviewed 29 August 2026
Sum of the first n terms of a geometric series and the convergence condition of an infinite series.
- Wybrane wzory matematyczne na egzamin maturalny z matematyki
Centralna Komisja Egzaminacyjna · Reviewed 29 August 2026
Official Polish examination reference for geometric sequences and convergent series.