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Geometric Progression Solver - Free Online Calculator

Solve geometric progressions online. Calculate the nth term, sum, common ratio, and generate sequences. Free geometric sequence calculator.

What is Geometric Progression Solver?

A geometric progression (GP), also called a geometric sequence, is a sequence 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. For example, 2, 6, 18, 54, 162 is a geometric progression with first term a = 2 and common ratio r = 3. Geometric progressions appear everywhere in mathematics, science, finance, and nature. Compound interest follows a geometric progression — if you invest $1,000 at 5% annual interest, your balance after each year forms the sequence 1000, 1050, 1102.50, 1157.63, and so on (ratio = 1.05). Population growth, radioactive decay, sound frequency octaves, and fractal patterns all follow geometric progressions. This solver handles all standard GP calculations: finding any term in the sequence, computing partial sums, determining the sum to infinity for convergent series (|r| < 1), and generating the full sequence for visualization.

How to Use Geometric Progression Solver

Enter the first term (a) and common ratio (r) of your geometric progression. Optionally enter the number of terms (n) to generate. The solver instantly calculates the nth term, the sum of n terms, and the sum to infinity (when applicable). The full sequence is displayed below the results. All formulas used are shown alongside the answers for educational reference.

How Geometric Progression Solver Works

The solver uses the standard geometric progression formulas: • nth term: aₙ = a × r^(n-1) Example: For a=2, r=3, n=5: a₅ = 2 × 3⁴ = 2 × 81 = 162 • Sum of n terms: Sₙ = a × (rⁿ - 1) / (r - 1) when r ≠ 1 Example: For a=2, r=3, n=5: S₅ = 2 × (243 - 1) / 2 = 242 • Sum to infinity: S∞ = a / (1 - r) when |r| < 1 Example: For a=10, r=0.5: S∞ = 10 / 0.5 = 20 The solver automatically detects whether the infinite sum exists (only when the absolute value of the common ratio is less than 1) and displays it when applicable. For |r| ≥ 1, the series diverges and no infinite sum is shown.

Common Use Cases

  • Solving homework and exam problems in algebra, pre-calculus, and sequences & series courses
  • Calculating compound interest growth over multiple periods in financial planning
  • Modeling population growth or decay (bacteria doubling, radioactive half-life)
  • Computing signal attenuation in engineering (each stage reduces signal by a constant factor)
  • Checking hand calculations for GP problems in standardized tests (SAT, GRE, GMAT)
  • Exploring mathematical patterns and visualizing how sequences grow or converge

Frequently Asked Questions

What is a geometric progression?▼

A geometric progression (GP) is a sequence where each term is found by multiplying the previous term by a fixed number called the common ratio. For example: 2, 6, 18, 54 has ratio = 3. Each term is 3 times the previous one.

What formulas does this solver use?▼

It uses three standard GP formulas: nth term = a × r^(n-1), Sum of n terms = a × (r^n - 1) / (r - 1) when r ≠ 1, and Sum to infinity = a / (1 - r) when |r| < 1. All formulas are displayed alongside results.

When does the sum to infinity exist?▼

The sum to infinity exists only when the absolute value of the common ratio is less than 1 (|r| < 1). This means the terms get progressively smaller and the sum converges to a finite value. For example, 1 + 0.5 + 0.25 + 0.125 + ... converges to 2.

Can the common ratio be negative?▼

Yes. A negative common ratio creates an alternating sequence where terms switch between positive and negative. For example, a=1, r=-2 gives 1, -2, 4, -8, 16, -32, and so on.

Can the common ratio be a decimal or fraction?▼

Yes. The common ratio can be any non-zero number including decimals and fractions. For example, r=0.5 creates a halving sequence (10, 5, 2.5, 1.25, ...) and r=1.05 models 5% compound growth.

How many terms can the solver generate?▼

The solver can calculate and display sequences of any reasonable length. For very large values of n or r, the numbers grow extremely fast (exponential growth), so the display focuses on the most useful information.

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