Geometric Sequence Calculator
Calculate the nth term, sum, and first 10 terms of a geometric sequence given the first term and common ratio.
nth Term (aₙ)
—
Sum (Sₙ)
—
First 10 Terms
What is Geometric Sequence Calculator?
A geometric sequence (geometric progression) is a sequence where each term is obtained by multiplying the previous term by a fixed number called the common ratio. Geometric sequences model exponential growth and decay, compound interest, population growth, and many natural phenomena.
How to use
- 1 Enter the first term (a₁) of the sequence.
- 2 Enter the common ratio (r) — greater than 1 for growth, between 0 and 1 for decay, negative for alternating.
- 3 Enter n — the term number you want to calculate.
- 4 The nth term, sum of first n terms, and the first 10 terms appear instantly.
Formula
Example calculation
For a₁ = 3, r = 2, n = 6: a₆ = 3 × 2⁵ = 96. S₆ = 3 × (1−64)/(1−2) = 3 × 63 = 189. First 10 terms: 3, 6, 12, 24, 48, 96, 192, 384, 768, 1536.
Frequently asked questions
What happens when r is between 0 and 1?
The terms decrease toward zero. For example, a₁=100, r=0.5 gives 100, 50, 25, 12.5, … — this models radioactive decay or depreciating assets.
What is an infinite geometric series?
When |r| < 1, the sum of infinitely many terms converges: S∞ = a₁ / (1−r). This is not computed here, but for a₁=1, r=0.5: S∞ = 2.
Can r be negative?
Yes. A negative r creates alternating sequences. For example, a₁=1, r=−2 gives 1, −2, 4, −8, 16, …
How does compound interest relate to geometric sequences?
Compound interest is a geometric sequence where a₁ = principal, r = (1 + rate), and each term is the balance after one compounding period.
What is the common ratio if I know two consecutive terms?
r = aₙ₊₁ / aₙ. Divide any term by the previous term to find the common ratio.