Math Tools
Geometric Sequence Calculator
Find the nth term, generate terms, and calculate geometric sequence sums.
Geometric Sequence Calculator
Find the nth term, generate the sequence, and calculate the sum of a geometric sequence.
Formulas
aₙ = a × r^(n − 1)
Sₙ = a(1 − rⁿ) / (1 − r)
Results
Nth term
96
Sum of first n terms
189
Sequence
3, 6, 12, 24, 48, 96
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What this calculator does
A geometric sequence changes by multiplying each term by a constant ratio. This calculator finds the nth term, generates the sequence, and calculates the sum of the first n terms.
What is a geometric sequence?
A geometric sequence is a number pattern where each term is found by multiplying the previous term by the same constant number. That constant number is called the common ratio. For example, in the sequence 3, 6, 12, 24, 48, each term is multiplied by 2.
Geometric sequences appear in algebra, finance, population growth, physics, computer science, and compound interest problems. They are important because repeated multiplication creates a predictable pattern that can be described with formulas.
Geometric sequence formulas
Nth term formula
aₙ = a × r^(n − 1)
Use this formula to find any term in the sequence when you know the first term, common ratio, and term number.
Sum of n terms formula
Sₙ = a(1 − rⁿ) / (1 − r)
Use this formula to calculate the total of the first n terms in a geometric sequence when the common ratio is not equal to 1.
Geometric sequence term table
This table shows the first terms of the current geometric sequence input.
| Term number | Term value |
|---|---|
| 1 | 3 |
| 2 | 6 |
| 3 | 12 |
| 4 | 24 |
| 5 | 48 |
| 6 | 96 |
How to find the nth term of a geometric sequence
To find the nth term of a geometric sequence, start with the first term, identify the common ratio, and choose the term number you want to find. Then apply the formula aₙ = a × r^(n − 1).
For example, if the first term is 3, the common ratio is 2, and you want the 6th term, the result is 3 × 2^(6 − 1) = 3 × 32 = 96. This calculator does that automatically and also generates the earlier terms so you can check the pattern.
How to find the sum of a geometric sequence
The sum of a geometric sequence means adding the first n terms together. Instead of adding them one by one, you can use the geometric series sum formula Sₙ = a(1 − rⁿ) / (1 − r).
This is useful for series problems, finance calculations, repeated multiplication models, and classroom algebra. It becomes especially helpful when the sequence has many terms and manual addition would take too long.
Geometric sequence example
Suppose the first term is 3, the common ratio is 2, and the number of terms is 6. The sequence is 3, 6, 12, 24, 48, 96.
The 6th term is 96. The sum of the first 6 terms is 3 + 6 + 12 + 24 + 48 + 96 = 189. This example shows how the sequence grows by repeated multiplication rather than repeated addition.
Where geometric sequences are used
Geometric sequences are used in compound interest, depreciation, population growth, signal processing, computer science, and algebra. Any pattern that changes by multiplying by the same factor can often be modeled as a geometric sequence.
Students also use geometric sequence formulas in exams and homework to solve sequence and series problems quickly. An online geometric sequence calculator makes it easier to verify answers and understand how the formulas work.
Geometric sequence calculator FAQ and common questions
What is a geometric sequence?
A geometric sequence is a sequence where each term is found by multiplying the previous term by the same constant ratio.
What is the formula for the nth term of a geometric sequence?
The nth term formula is aₙ = a × r^(n − 1), where a is the first term, r is the common ratio, and n is the term number.
How do you find the sum of a geometric sequence?
Use the formula Sₙ = a(1 − rⁿ) / (1 − r) when the common ratio is not equal to 1.
What is the common ratio in a geometric sequence?
The common ratio is the fixed value used to multiply one term to get the next term in the sequence.
What is the difference between an arithmetic sequence and a geometric sequence?
An arithmetic sequence changes by adding or subtracting the same amount each time, while a geometric sequence changes by multiplying by the same ratio each time.
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