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First term of the sequence

Common ratio

N

N-th term of the sequence

Sum of first N terms

Infinite sum

This geometric sequence calculator will help you to analyze a geometric sequence - that is, a sequence in which each term is constructed by multiplying the previous one by a constant factor. You will be able to find this factor, as well as any term of the sequence and the geometric series - sum of n consecutive terms. If the sequence has a sum to infinity, our geometric sum calculator will find it as well. If you want to know what rules and formulas does it use, simple keep reading!

A geometric sequence is an ordered set of numbers, in which each consecutive number is found by multiplying the previous term by a factor called the **common ratio**. Just as in case of any other sequence, it can have a finite (for example 30) or an infinite number of terms.

Geometric sequences are uniquely defined by two main coefficients: the **common ratio** and the **first term**. If you know these two values, you will be able to calculate each term of the sequence in a few seconds.

Some examples of a geometric sequence include:

- 2, 4, 8, 16, 32, 64, 128...
- 6, -3, 1.5, -0.75, 0.375...
- 1, -1, 1, -1, 1, -1, 1... (called the alternating sequence with
`r = -1`

)

What happens if you want to calculate a certain term of a geometric sequence - for example, the tenth one? You surely don't want to keep multiplying the numbers over and over. You probably have already noticed that you don't have to. The only thing you need to do is to take the first term and multiply it by the common ratio raised to a certain power. When put into a formula, this rule looks like this:

`a = a₁ * rⁿ⁻¹`

where:

**a**is the nᵗʰ term of the sequence,**r**is the common ratio and**a₁**is the first term of the sequence.

Now, let's assume that you want to calculate the geometric series with our geometric sequence calculator (a series is simply a sum of the sequence). Do you need to calculate each and every term and finally add them together?

The answer is no. You can use a geometric sum equation that requires you to know only three things - the common ratio, the first term and number of terms you want to add up. This formula is derived using the properties of polynomial division.

`Σ = a₁ * (1 - rⁿ) / (1-r)`

You probably noticed that if the common ratio is lower than 1, each consecutive term is smaller than the previous one. That means that at some point the terms will become so small that they will virtually be equal to 0 and not have any influence on the series. Such sequences have an **infinite sum** - the sum of all terms, from the first one to infinity.

The necessary condition for a geometric sequence to have an infinite sum is that the absolute value of the common ratio must be lower than 1 (`|r| < 1`

).

Then, you can find the infinite series according to the formula

`Σ = a₁ / (1-r)`

Our geometric sequence calculator is capable of analyzing only one specific type of a sequence. For example, the sequence 3, 5, 7, 9, 11, 13, 15... does not have a common ratio, so its sum cannot be found using this tool. Feel free to check out our arithmetic sequence calculator if you want to solve that sequence.