Geometric Sequence Calculator — Everything About Geometric Progressions
Our Geometric Sequence Calculator helps you explore any geometric progression: find the nth term, calculate the partial sum, determine the infinite sum, and list all sequence terms side by side. Whether you are studying math, working with compound interest, or modeling exponential growth, this tool has you covered.
What Is a Geometric Sequence?
A geometric sequence (also called a geometric progression) is a list of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio (r). The defining property is that the ratio of any two consecutive terms is constant:
aₙ / aₙ₋₁ = r (for all n > 1)For example, the sequence 2, 6, 18, 54, 162, … is geometric with first term a₁ = 2 and common ratio r = 3.
Explicit Formula for a Geometric Sequence
The explicit formula allows you to find any term directly without knowing the previous one:
aₙ = a₁ × r^(n−1)Where:
- a₁ — the first term
- r — the common ratio
- n — the position (index) of the term
Recursive Formula for a Geometric Sequence
The recursive formula defines each term in relation to the previous one:
a₁ = (given), aₙ = aₙ₋₁ × rBoth formulas describe the same sequence; the explicit form is more convenient for direct calculation, while the recursive form shows the underlying step-by-step structure.
Geometric Series Formula: Sum of a Geometric Sequence
The sum of the first n terms of a geometric sequence (called a geometric series) is:
Sₙ = a₁ × (1 − rⁿ) / (1 − r), when r ≠ 1
Sₙ = n × a₁, when r = 1Infinite Sum of a Geometric Series
When the absolute value of the common ratio is less than 1 (|r| < 1), the geometric series converges to a finite sum as n → ∞:
S∞ = a₁ / (1 − r), valid only when |r| < 1If |r| ≥ 1, the series diverges — it grows without bound or oscillates, and no finite infinite sum exists.
Convergent vs. Divergent Geometric Sequences
- |r| < 1 — convergent; terms approach 0; infinite sum exists
- |r| = 1 — constant sequence (r = 1) or alternating ±a₁ (r = −1)
- |r| > 1 — divergent; terms grow without bound
How to Use This Geometric Sequence Calculator
- Enter the First Term (a₁) — the starting value of your sequence.
- Enter the Common Ratio (r) — the multiplier applied at each step. Cannot be zero.
- Choose the Number of Terms (n) — how many terms to display (1 to 20).
- Optionally select a Unit or Currency:
- Currency: USD, EUR, GBP, JPY, CNY, INR, RUB, BRL, AUD, CAD, and more
- Metric: m, cm, km, kg, L, etc.
- Imperial (US): ft, in, mi, lb, gal, etc.
- Click Calculate Sequence to see the nth term, partial sum, infinite sum, and the full term table.
Geometric Sequence Examples
Example 1: Simple Growth (r = 3)
a₁ = 1, r = 3, n = 5
Terms: 1, 3, 9, 27, 81 | a₅ = 81 | S₅ = 121
Example 2: Halving (r = 0.5, converges)
a₁ = 64, r = 0.5, n = 6
Terms: 64, 32, 16, 8, 4, 2 | S₆ = 126 | S∞ = 128
Example 3: Financial Growth (monthly investment)
If an investment doubles every period (r = 2) starting at $1,000:
$1,000 → $2,000 → $4,000 → $8,000 → … | a₁₀ = $512,000
Example 4: Zeno's Paradox
Zeno argued that an arrow can never reach its target because it must first cover half the distance, then half of the remaining, and so on forever. This gives the geometric series:
1/2 + 1/4 + 1/8 + … = S∞ = (1/2) / (1 − 1/2) = 1The infinite sum is 1 — the arrow does reach the target!
FAQs
What is the difference between arithmetic and geometric sequences?
In an arithmetic sequence, each term is obtained by adding a constant (the common difference). In a geometric sequence, each term is obtained by multiplying by a constant (the common ratio).
Can the common ratio be negative?
Yes. A negative ratio produces an alternating sequence (positive, negative, positive, …). For example: 3, −6, 12, −24 with r = −2.
Can the first term be negative?
Yes. The formulas work for any non-zero first term, positive or negative.
What happens when r = 1?
Every term equals a₁, and the sum Sₙ = n × a₁.
How is a geometric sequence used in finance?
Compound interest produces geometric growth: each period the balance is multiplied by (1 + interest rate), making balances a geometric sequence. Loan amortization, population growth, and radioactive decay also follow geometric patterns.