Calculadora de Progressão Geométrica
- Created by
- Renato Passos, Eng. de Software
- Reviewed by
- Renato Passos, Eng. de Software
Last updated: Apr 18, 2026
Formula
aₙ = a₁ · rⁿ⁻¹ ; Sₙ = a₁ · (1 − rⁿ) / (1 − r)
About this calculator
This calculator solves Geometric Progression (GP) problems. You can find any term of the sequence using the formula aₙ = a₁ · rⁿ⁻¹, where a₁ is the first term, r is the common ratio, and n is the term position. It also calculates the sum of the first n terms with Sₙ = a₁ · (1 − rⁿ) / (1 − r), for r ≠ 1. Just enter the known values and the result appears instantly.
The tool is useful for students and professionals dealing with geometric sequences in math, physics, or finance. For example, compound interest, installment amounts, or exponential population growth naturally involve GP. Knowing the general term or partial sum helps plan investments or understand growth phenomena.
Be careful when using the sum formula if the ratio r equals 1: in that case, the GP is constant and the sum is simply n · a₁. Also, ensure n is a positive integer. The calculator assumes r ≠ 1 for the sum but shows a warning if r = 1. Always check whether the problem asks for the general term or the cumulative sum.
Frequently asked questions
What is a Geometric Progression?
It is a sequence where each term after the first is the previous term multiplied by a constant called the common ratio (r). Example: 2, 6, 18, 54 (ratio 3).
How do I calculate the general term of a GP?
Use the formula aₙ = a₁ · rⁿ⁻¹. You need the first term (a₁), the common ratio (r), and the term position (n).
Does the sum formula work for any ratio?
It works for r ≠ 1. If r = 1, the GP is constant and the sum is n · a₁. The calculator warns if r = 1.
Can I use this calculator for compound interest?
Yes, compound interest follows a GP: the amount after n periods is M = P · (1 + i)ⁿ, which is the general term with a₁ = P and r = 1 + i.
What if n is not an integer?
GP is defined for positive integer n (term position). If n is not an integer, the result is meaningless. Enter only integers greater than zero.