Geometric Series Calculator
Sum of n Terms (Sₙ)80
nth Term (aₙ)54
Infinite Sum (|r| < 1)Diverges (undefined)
The Geometric Series Calculator multiplies a starting value by a fixed ratio over and over, then reports the nth term, the sum of a finite run of terms, and — when the series converges — the sum it would reach if continued forever. Geometric growth shows up in compound interest, population models, and repeating fractions, where each step scales the previous one rather than adding a constant.
Formula
aₙ = a₁ · r^(n−1) ; Sₙ = a₁(1 − rⁿ) / (1 − r) ; S∞ = a₁ / (1 − r) for |r| < 1
- a₁
- First term of the sequence
- r
- Common ratio between consecutive terms
- n
- Number of terms in the finite sum
- aₙ
- Value of the nth term
- S∞
- Sum to infinity, defined only when |r| < 1
How it works
- Enter the first term a₁, the common ratio r (the factor each term is multiplied by), and the number of terms n.
- The nth term comes from aₙ = a₁ · r^(n−1), and the finite sum uses Sₙ = a₁(1 − rⁿ) / (1 − r), with the special case Sₙ = a₁·n when r equals 1.
- If the ratio satisfies |r| < 1 the series converges and the infinite sum a₁ / (1 − r) is reported; otherwise the infinite sum is undefined because the terms do not shrink toward zero.
Worked example
Find the 4th term and the sum of the first 4 terms with a₁ = 2 and r = 3.
- nth term: a₄ = 2 × 3^(4−1) = 2 × 27 = 54.
- Finite sum: S₄ = 2(1 − 3⁴) / (1 − 3) = 2(1 − 81) / (−2) = 2(−80)/(−2) = 80.
- Because |3| ≥ 1, the infinite sum diverges and is undefined.
The 4th term is 54 and the sum of the first 4 terms is 80; the infinite sum diverges.
Frequently asked questions
- When does a geometric series have an infinite sum?
- Only when the common ratio satisfies |r| < 1. In that case successive terms shrink toward zero and the partial sums settle on a finite limit equal to a₁ / (1 − r); otherwise the sum grows without bound.
- What happens when the common ratio equals 1?
- Every term equals the first term, so the standard formula divides by zero and is skipped. The finite sum is simply a₁ multiplied by the number of terms, and there is no finite infinite sum.
- How is a geometric series different from an arithmetic series?
- A geometric series multiplies by a constant ratio at each step, while an arithmetic series adds a constant difference. That multiplicative growth is why a geometric series can converge to a finite total when the ratio is small.
- Can the first term or ratio be negative?
- Yes. A negative ratio creates an alternating series whose terms switch sign, and it still converges to a finite infinite sum as long as the ratio is between −1 and 1 in value.