Termo Geral PG
- Created by
- Renato Passos, Eng. de Software
- Reviewed by
- Renato Passos, Eng. de Software
Last updated: Apr 18, 2026
Formula
aₙ = a₁·rⁿ⁻¹
About this calculator
The General Term of a Geometric Progression (GP) calculator allows you to find any term of the sequence given the first term (a₁) and the common ratio (r). The formula used is aₙ = a₁ · r^(n−1), where n is the position of the desired term. This tool is useful for students and professionals working with numerical sequences, such as in financial mathematics, physics, or computer science.
To use the calculator, enter the first term (a₁), the common ratio (r), and the term index (n). The result is the value of the term at position n. For example, for a GP with a₁ = 2 and r = 3, the term at position 4 is 2 · 3^(4−1) = 2 · 27 = 54. The calculator handles real numbers, including decimals and negatives.
When to use? Whenever you need to determine a specific term of a geometric progression, such as calculating the amount of an investment with compound interest (which follows a GP), or predicting populations in exponential growth. It is also useful for solving sequence problems in exams and competitions.
Cautions: verify that the common ratio (r) is not zero (if r=0, all terms from the second onward are zero). For very large indices, the result may be extremely large or small, causing overflow in some calculators. Remember that the formula assumes the sequence starts at n=1.
Frequently asked questions
What is a geometric progression?
A geometric progression (GP) is a numerical sequence where each term after the first is obtained by multiplying the previous term by a constant called the common ratio (r).
How to calculate the general term of a GP?
Use the formula aₙ = a₁ · r^(n−1), where a₁ is the first term, r is the common ratio, and n is the position of the desired term.
Can I use the calculator for fractional ratios?
Yes, the calculator accepts decimal numbers, including fractions represented as decimals. Just enter the ratio value in decimal format.
What happens if the ratio is negative?
The GP will have alternating signs. The calculator handles negative ratios normally, as long as the exponent is an integer.
Does the calculator work for very large n?
Yes, but the result may be very large or very small. Depending on the browser, overflow or underflow may occur.