Geometric Sequence Calculator
Find the nth term, the sum of the first n terms and the infinite sum of a geometric sequence.
nth term
1,536
- Sum of the first n terms (Sₙ)
- 3,069
- First terms
- 3, 6, 12, 24, 48, 96, 192, 384, …
- Infinite series
- Does not converge, since |r| is 1 or more
The sequence has 10 terms in the calculation.
Calculation breakdown
| Step | Formula | Result |
|---|---|---|
| nth term | 3 × 2 ^ (10 − 1) | 1,536 |
| Sum of the first n terms | 3 × (1 − 2 ^ 10) ÷ (1 − 2) | 3,069 |
How it works
Enter the first term, the ratio between terms and how many terms to use.
Worked example
First term 3, ratio 2, 10 terms:
- Terms: 3, 6, 12, 24, 48, 96, 192, 384, …
- 10th term: 3 × 2⁹ = 1,536
- Sum of 10 terms: 3 × (1 − 2¹⁰) ÷ (1 − 2) = 3,069
Sources
Frequently asked questions
What is a geometric sequence?
A geometric sequence is a list of numbers where each term is the previous one times the same number, the common ratio. 3, 6, 12, 24 has a ratio of 2.
What is the formula for the nth term?
aₙ = a₁ × rⁿ⁻¹. For a first term of 3 and a ratio of 2, the 10th term is 3 × 2⁹ = 1,536.
How do you find the sum of a geometric sequence?
The sum of the first n terms is a₁ × (1 − rⁿ) ÷ (1 − r), when r is not 1. For 3, 6, 12 and so on for 10 terms, the sum is 3 × (1 − 1,024) ÷ (1 − 2) = 3,069.
When does an infinite geometric series have a sum?
When the ratio is between −1 and 1. The sum is then a₁ ÷ (1 − r). The series 1 + 1/2 + 1/4 + … adds up to 2. If |r| is 1 or more, the terms do not shrink and the series has no finite sum.
Can the ratio be negative?
Yes. The terms then alternate in sign, like 2, −6, 18, −54 with a ratio of −3.