Geometric Sequence Calculator
Find any term and the sum of a geometric sequence from the first term and ratio.
Enter the first term, the common ratio, and the number of terms to compute the nth term, the sum of the terms, and the infinite sum.
nth term (aₙ):
Sum of n terms (Sₙ):
Infinite sum (S∞):
First terms:
Geometric Sequence vs. Geometric Series
A geometric sequence is an ordered list of numbers in which each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio (r). For example, 2, 6, 18, 54 is a geometric sequence with first term 2 and common ratio 3.
A geometric series is the sum of the terms of a geometric sequence. While the sequence lists the individual values, the series adds them together, for example 2 + 6 + 18 + 54 = 80.
The Formulas
- nth term: aₙ = a₁ · r^(n−1)
- Sum of n terms (r ≠ 1): Sₙ = a₁ · (1 − rⁿ) / (1 − r)
- Sum of n terms (r = 1): Sₙ = a₁ · n
- Infinite sum (|r| < 1): S∞ = a₁ / (1 − r)
Example: With a₁ = 2, r = 3, n = 4, the terms are 2, 6, 18, 54. The 4th term is 54 and the sum S₄ = 80.
Convergence
An infinite geometric series converges to a finite sum only when the absolute value of the common ratio is less than 1, that is when |r| < 1. In that case S∞ = a₁ / (1 − r). When |r| ≥ 1, the terms do not shrink toward zero, the partial sums grow without bound, and the infinite series diverges (it has no finite sum).
Related Calculators
Note: This calculator provides mathematical calculations for geometric sequences and series based on the formulas described. While we strive for accuracy, please verify important calculations independently. This tool is for educational and informational purposes and should not be the sole basis for financial, academic, or professional decisions.