Geometric sequence calculator
Enter the first term, the common ratio and the number of terms. You get the nth term, the sum of the first n terms, the sum to infinity when it exists, and a list of the first terms.
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How to use it
- Enter the first term and the common ratio.
- Enter how many terms you want to add up or the position of the term you need.
- Read the nth term and the sums. A list of up to the first 10 terms is shown for checking.
How it works
The nth term is a·r^(n−1). The partial sum is S = a(1 − r^n)/(1 − r), or a·n if r = 1. If |r| < 1 the infinite sum is a/(1 − r). Results are shown with up to 12 significant digits.
Worked example
Take a = 1000, r = 0.5 and n = 4.
- Terms: 1000, 500, 250, 125
- 4th term: 1000 × 0.5³ = 125
- Sum of the 4 terms: 1000 × (1 − 0.0625) / 0.5 = 1875
- Sum to infinity: 1000 / (1 − 0.5) = 2000
Tips
- A ratio below 0 makes the terms alternate in sign.
- A ratio of exactly 1 gives a constant sequence; −1 flips between a and −a.
- Compound interest follows a geometric sequence with r = 1 + rate.
Frequently asked questions
What is a geometric sequence?
A sequence where each term is the previous one multiplied by the same number, the common ratio r. For example 3, 6, 12, 24 has a first term of 3 and a ratio of 2.
What is the formula for the nth term?
The nth term is a × r^(n−1), where a is the first term. With a = 3 and r = 2 the 6th term is 3 × 2^5 = 96.
How is the sum of a geometric sequence calculated?
The sum of the first n terms is a × (1 − r^n) / (1 − r) when r is not 1. When r equals 1 every term is a, so the sum is a × n.
When does a geometric series have a sum to infinity?
Only when the absolute value of r is less than 1, so the terms shrink toward 0. Then the total approaches a / (1 − r). With a = 1000 and r = 0.5 that is 2000.
How do I find the common ratio?
Divide any term by the one before it, for example the second term by the first. If the ratio is the same all along the list, the sequence is geometric.