What is a Geometric Sequence?

Learn more about geometric sequences so you can better interpret the results provided by this calculator: A geometric sequence, also known as a geometric progression, is a sequence of numbers \(a_1, a_2, a_3, ....\) with the specific property that the ratio between two consecutive terms of the sequence is ALWAYS constant, equal to a certain value \(r\).

One way of completely determining a geometric sequence is by knowing its starting point \(a_1\) and the common ratio \(r\), but that is not the only way.

Using this Geometric Sequence Calculator

In order to use this calculator, you need to simply provide the initial value of the sequence \(a_0\), and the constant ratio \(r\), and then click on "Calculate", to get the steps shown.

You also need to provide the number of steps \(n\) you want to add. If you want to add an infinite number of terms, use this geometric series calculator.

Geometric Sequence Formula

The value of the \(n^{th}\) term of the arithmetic sequence, \(a_n\) is computed by using the following formula:

\[a_n = a_1 r^{n-1}\]

The above formula allows you to find the find the nth term of the geometric sequence. This means that in order to get the next element in the sequence we multiply the ratio \(r\) by the previous element in the sequence.

So then, the first element is \(a_1\), the next one is \(a_1 r\), the next one is \(a_1 r^2\), and so on.

Notice that a geometric series is defined by the recurrent formula \(a_{n+1} = r a_n \), which can be solved inductively to give the explicit formula for the geometric sequence which is shown above.

It is a explicit formula in the sense that it tells you exactly how to get \(a_n\) as a function of \(a_0\), \(n\) and \(r\), this is in terms of the initial value, number of steps and common ratio.

Geometric and Arithmetic Sequences: How do they differ

For this type of sequence, the ratio between two consecutive values in the sequence is constant. If you are dealing with the case in which the difference between any two consecutive values of the sequence is constant, then you use use our arithmetic sequence calculator instead.

On the other hand, if you want to add an infinite geometric series, you can use this geometric series calculator .

Common Ratio Calculator

Sometimes this geometric sequence calculator is referred to as a common ratio calculator and for good reason, considering that all consecutive terms in a geometric sequence have a common ratio.

It is indeed important for you to know the different 'jargon' used when referring to this type of calculator. In Algebra and Calculus there are many types of sequences and series, and the geometric sequences are ones that play a special role in many applications.

One that comes to mind, for example, is the Fibonacci sequence, which unlike this one, has an additive construction, as opposed to being multiplicative like the one used for geometric sequences.