Geometric Sequence Calculator
Find the nth term, finite sum, and infinite sum of any geometric progression instantly. Get step-by-step solutions and sequence generation.
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Calculation Results
Nth Term (aₙ)
Finite Sum (Sₙ)
Infinite Sum (S∞)
Common Ratio (r)
Number of Terms (n)
Generated Sequence
Step-by-Step Solution
Formula & Explanation
What is a geometric sequence?
A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.
What is a common ratio (r)?
The common ratio is the amount you multiply by each time. You can find it by dividing any term by the previous term: r = a₂ / a₁.
Nth Term Formula
aₙ = a × r^(n − 1)
Where a is the first term, r is the common ratio, and n is the position of the term.
Finite Sum Formula (Sₙ)
Sₙ = a × (1 − rⁿ) / (1 − r)
Used to find the sum of the first n terms when r ≠ 1. If r = 1, the formula is simply Sₙ = n × a.
Infinite Sum Formula (S∞)
S∞ = a / (1 − r)
An infinite geometric series only converges (has a finite sum) if the absolute value of the common ratio is strictly less than 1 (|r| < 1).
Frequently Asked Questions
It is an ordered list of numbers where each term is found by multiplying the previous term by a constant value, known as the common ratio.
You use the formula aₙ = a × r^(n-1). Just plug in your first term (a), common ratio (r), and the position number (n) to find the value.
The common ratio is the constant multiplier between consecutive terms in a geometric sequence. It can be positive, negative, integer, or a fraction.
For a finite sequence, use Sₙ = a(1 - rⁿ) / (1 - r). This calculates the total sum of the first 'n' terms.
An infinite series converges only if the absolute value of the common ratio is less than 1 (|r| < 1). If |r| ≥ 1, the series diverges and the infinite sum cannot be calculated.
Yes. When the common ratio is negative, the signs of the terms in the sequence will alternate between positive and negative.
If the ratio is 1, the sequence is just the first term repeating infinitely (e.g., 5, 5, 5, 5). The nth term is simply 'a', and the finite sum is 'n × a'.
Absolutely. A geometric sequence can start with any number, positive or negative (except 0, which would just yield a sequence of zeros).