Geometric Sequence Calculator
Enter the first term, the common ratio, and how many terms you want. The calculator lists every term, the nth term aₙ, the partial sum Sₙ, and — when |r| < 1 — the infinite sum S∞.
nth term (a₁₀)
1536
- Sum of 10 terms (Sₙ)
- 3069
- First term (a₁)
- 3
- Terms
- 3, 6, 12, 24, 48, 96, 192, 384, 768, 1536
aₙ = a₁ × r^(n − 1). For r ≠ 1, Sₙ = a₁ × (1 − rⁿ) / (1 − r). When |r| < 1 the series converges to S∞ = a₁ / (1 − r). Source: OpenStax Algebra & Trigonometry, ch. 9.
How to use this calculator
Type the first term a₁ (any real number), the common ratio r (the multiplier between consecutive terms — use a fraction like 0.5 for a shrinking sequence, a negative ratio for an alternating sequence), and the number of terms n (a whole number from 1 to 1000). The calculator prints the full list of terms, the value of the nth term, the partial sum Sₙ, and — whenever |r| < 1 — the infinite sum S∞ the series converges to.
How the calculation works
A geometric sequence is a list of numbers in which each term is the previous one multiplied by the same constant r. The closed-form formula for the nth term is aₙ = a₁ × r^(n − 1). The partial sum of the first n terms is Sₙ = a₁ × (1 − rⁿ) / (1 − r) whenever r ≠ 1, and simply n × a₁ when r = 1. If the absolute value of r is strictly less than 1, the partial sums approach a finite limit as n grows — that limit is the infinite sum S∞ = a₁ / (1 − r). The calculator computes every term and the closed-form sum in one pass, so even 1,000-term sequences resolve instantly. Source: OpenStax Algebra & Trigonometry, ch. 9.
Worked example
Take a₁ = 3 and r = 2 with n = 5 terms. The sequence is 3, 6, 12, 24, 48. The fifth term is a₅ = 3 × 2⁴ = 3 × 16 = 48. The sum is S₅ = 3 × (1 − 32) / (1 − 2) = 3 × (−31) / (−1) = 93 — matching the direct addition 3 + 6 + 12 + 24 + 48 = 93. Since |r| = 2 ≥ 1, this series diverges and has no infinite sum. A converging example: a₁ = 1, r = ½ gives the famous series 1 + ½ + ¼ + ⅛ + … whose infinite sum is S∞ = 1 / (1 − ½) = 2.
Frequently asked questions
What is the difference between a geometric sequence and an arithmetic sequence?
A geometric sequence has a constant ratio between terms — you multiply by the same amount each step. An arithmetic sequence has a constant difference — you add the same amount each step. So 3, 6, 12, 24 is geometric (r = 2), but 3, 7, 11, 15 is arithmetic (d = 4). If you need the arithmetic version, use the arithmetic sequence calculator instead.
When does the infinite sum exist?
Only when the absolute value of the common ratio is strictly less than 1 (i.e. −1 < r < 1 and r ≠ 0). In that case the terms shrink toward zero fast enough that the running total approaches a finite limit, S∞ = a₁ / (1 − r). If |r| ≥ 1 the terms either stay the same size or grow, so the partial sums march off to infinity (or oscillate without settling) and there is no infinite sum.
What is the formula for the nth term?
aₙ = a₁ × r^(n − 1). The exponent is (n − 1), not n, because the first term has been multiplied by r zero times — it is just a₁. By the time you reach the nth term you have multiplied by r exactly (n − 1) times.
How is the partial sum formula derived?
Write Sₙ = a₁ + a₁r + a₁r² + … + a₁r^(n−1). Multiply both sides by r: r Sₙ = a₁r + a₁r² + … + a₁rⁿ. Subtract the second equation from the first: Sₙ − r Sₙ = a₁ − a₁rⁿ, so Sₙ (1 − r) = a₁ (1 − rⁿ). Dividing by (1 − r) gives Sₙ = a₁ × (1 − rⁿ) / (1 − r), valid whenever r ≠ 1. When r = 1 every term is a₁, so the sum is simply n × a₁.
Can the common ratio be negative or a fraction?
Yes. A negative ratio produces an alternating sequence: a₁ = 5, r = −2 gives 5, −10, 20, −40, … A fractional ratio between 0 and 1 produces a shrinking sequence: a₁ = 64, r = 0.5 gives 64, 32, 16, 8, 4, … Both work fine in the formulas. The only ratio the calculator refuses is the value that would cause the formula to break — but even r = 1 is handled separately, so the only invalid input is non-numeric.
Is there a limit on the number of terms?
The calculator caps n at 1,000 to keep the page responsive — listing more than that becomes unreadable in a browser. For sequences longer than 1,000 terms, the nth term and the partial sum can still be computed by hand from the closed-form formulas above without enumerating every value: that is exactly the point of the closed-form sum.