How Geometric Progressions Work
A Geometric Progression (GP) is a sequence of numbers where each term is obtained by multiplying the previous one by a constant called the common ratio (r). For example: 2, 6, 18, 54... is a GP with a₁ = 2 and r = 3, because each term is the previous multiplied by 3.
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Geometric progressions have practical applications: compound interest, population growth, radioactive decay, loan installments, musical scales and much more. Our AI calculator allows you to instantly find any element of the GP.
GP Calculator Advantages
- Instant Calculation: Results in milliseconds with optimized algorithms and advanced AI processing.
- Multiple Calculations: Calculate general term, finite sum, infinite sum, ratio and first term in a single tool.
- Guaranteed Precision: Validated mathematical algorithms for correct results in any progression.
- Complete Visualization: See the generated sequence, GP graph and step-by-step calculation explanation.
- Works on Any Device: Responsive interface optimized for computers, tablets and smartphones.
- Free and No Registration: Use unlimited without registration, login or payment required.
Available Calculation Types
General Term
Find any term of the GP using aₙ = a₁ × r^(n-1).
Finite Sum
Calculate the sum of the first n terms using Sₙ = a₁(rⁿ - 1)/(r - 1).
Infinite Sum
For |r| < 1, calculate S∞ = a₁/(1 - r).
Find Ratio
Discover the ratio knowing terms of the sequence.
Interpolation
Insert geometric terms between two numbers.
Tips for Geometric Progressions
Memorize the Formulas
Term: aₙ = a₁ × r^(n-1). Finite sum: Sₙ = a₁(rⁿ - 1)/(r - 1). Infinite sum: a₁/(1-r).
Check the Ratio
In a GP, all quotients between consecutive terms must be equal.
Infinite Sum
Finite infinite sum only exists if |r| < 1. If |r| ≥ 1, the sum diverges.
Use the Graph
A GP graph is an exponential curve, not a straight line like in AP.
Geometric Mean
In 3 terms in GP, the middle one is the geometric mean: b = √(a × c).
Compound Interest
Amount with compound interest is a GP: M = C × (1 + i)ⁿ.