Geometric Sequence Calculator
Find any term, the partial sum, and the infinite sum of a geometric progression from its first term and common ratio.
Value of the term at the chosen index
- 1
Exponent (n − 1)
10 − 1 = 9 - 2
Ratio power r^(n−1)
0.5^9 = 0.002 - 3
nth term a_n = a1 × r^(n−1)
2 × 0.002 = 0.0039
How does this calculator work?
A geometric sequence multiplies each term by a common ratio r. The nth term is a1·r^(n−1), the sum of n terms is a1·(1−r^n)/(1−r) (or n·a1 when r=1), and the infinite sum is a1/(1−r) but only when |r| < 1; otherwise the series diverges.
Formula
How this is calculated
A geometric sequence multiplies each term by a fixed common ratio r. Enter the first term a1, the ratio r, and the term index n (a positive integer). The nth term is a_n = a1 · r^(n−1), so successive terms grow (|r| > 1), shrink (|r| < 1), or stay constant (|r| = 1), and alternate in sign when r is negative.
The finite sum of the first n terms uses S_n = a1 · (1 − r^n) / (1 − r) when r ≠ 1. When r = 1 every term equals a1, so the formula collapses to S_n = n · a1 and the calculator handles this case separately to avoid dividing by zero.
The infinite sum converges only when |r| < 1, giving S_∞ = a1 / (1 − r); otherwise the series diverges and no finite total exists, which the calculator reports instead of a number. Values are unitless, n must be a whole number ≥ 1, and the plotted curve shows the first terms (capped for very large n).
Frequently asked questions
Only when the absolute value of the common ratio is less than 1 (|r| < 1). Then the terms shrink toward zero fast enough to converge to a1 / (1 − r). Otherwise the series diverges.
Every term is identical to the first term, so the partial sum is simply n × a1. The standard formula would divide by zero, so this case is computed directly.
Yes. A negative ratio produces an alternating sequence whose signs flip each term. The infinite sum still converges as long as |r| < 1.
Also known as
TG we-Calculate Editorial Team. (2026). Geometric Sequence Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/geometric-sequence-calculator
TG we-Calculate Editorial Team. "Geometric Sequence Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/geometric-sequence-calculator.
TG we-Calculate Editorial Team, "Geometric Sequence Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/geometric-sequence-calculator
@misc{wecalculate_geometric_sequence_calculator, title = {Geometric Sequence Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/geometric-sequence-calculator}}, year = {2026}, note = {TG we-Calculate} }
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