Adblocker Detected

Uh Oh! It seems you’re using an Ad blocker!

We always struggled to serve you with the best online calculations, thus, there's a humble request to either disable the AD blocker or go with premium plans to use the AD-Free version for calculators.

Disable your Adblocker and refresh your web page 😊

Geometric Sequence Calculator

Geometric Sequence Calculator

Select the parameter and provide required entities. The tool will calculate geometric sequence, first term, common ratio, or number of terms against your selection.



First Term (a₁)

Common Ratio (r)

Number of Terms (n)

Enter n-th term a(n)

Sum of first n-terms S(n)




Table of Content

Get the Widget!

Add this calculator to your site and lets users to perform easy calculations.


How easy was it to use our calculator? Did you face any problem, tell us!

Calculate the geometric sequence, first term \(a_{1}\), common ratio (r), number of terms, and total sum with the geometric sequence calculator.

The geometric sequence solver provides step-wise calculations and graphs for a better understanding of the geometric series.

What Is the Geometric Sequence?

In mathematics, a geometric sequence is also called a geometric progression. It is defined as:

“A list of numbers in which each item in the Sequence is multiplied by a non-zero constant called the general ratio ‘r'”

When we multiply a constant (not zero) by the previous item, the next item in the Sequence appears.


\(a, a(r), a(r)^2, a(r)^3, a(r)^4, a(r)^5, …\)


a = first item 

 r =  common ratio.

Thus, the general term of a Geometric progression is given by \( ar^{n-1} \) and the general form of a Geometric sequence is \( a + a(r) + a(r)^2 + a(r)^3 + ….. \). The sum of geometric sequence calculator finds the nᵗʰ term and the sum of a geometric sequence from the first term to infinity.

How to Find the Sum of a Geometric Series?

Let \(a, ar, a(r)^2, a(r)^3, a(r)^4, ……ar^{n-1} \) is the given Geometric series.

Then the sum of finite geometric series is:

\(S_n = a + ar + a(r)^2 + a(r)^3 + a(r)^4 + …+ar^{n-1}\)

The formula to determine the sum of n terms of Geometric sequence is:

\(S_n = a[(r^n-1)/(r-1)] if r ≠ 1\)

Where a is the first item, n is the number of terms, and r is the common ratio.

Also, if the common ratio is 1, then the sum of the Geometric progression is given by:

\(S_{n} = na if r=1\)

The common ratio finder calculates the common ratio in the geometric sequence to determine all the terms.

Geometric Progression Formulas:

Below is a list of geometric progression formulas that can help to solve the various types of problems.

  • The general forms of GP terms are a, ar, a(r)^2, a(r)^3, a(r)^4, etc., where a is the first term and r is the common ratio.
  • The nth term of the Geometric sequence is k_n = ar^{n-1}
  • Common ratio = r = k_n/ k^{n-1}
  • The geometric sequence formula to determine the sum of the first n terms of a Geometric progression is given by:

S_n = a[(r^n-1)/(r-1)] if r > 1 and r ≠ 1

S_n = a[(1 – r^n)/(1 – r)] if r < 1 and r ≠ 1

  • The nth item at the end of GP, the last item is l, and the common ratio is r = l / [r (n – 1)].
  • The sum of infinite series, that is the sum of Geometric sequences with infinite terms is S∞ = a / (1-r) such that 1 >r >0.
  • If there are 3 values ​​in Geometric Progression, ​​then the middle one is known as the geometric mean of the other two items.
  • If a, b, and c are three values ​​in the Geometric Sequence, ​​then “b” is the geometric mean of “c” and “a”. This can be written as b = √ac or b^2 = ac
  • Assume that “r” and “a” are the common ratio and first term of a finite geometric sequence with n terms. Therefore, the kth item at the end of the geometric series will be ar^{n – k}.

The geometric sequence formula calculator computes the common ratio, the first and the nth terms of the geometric sequence.


Find the common ratio, where the first term \(a_1\) = 2 and \(a_3\) =16.


The common ratio calculator uses a simple formula for determining the ratio:

\(R = ^{n-1} \sqrt { a_n / a_1}\)

\(R = ^{3-1} \sqrt { 16 / 2}\)

\(R = ^{2} \sqrt { 8}\)

\(R = 2.82842712\)

How Does the Calculator Work?

The geometric calculator determines different geometric terms by following these steps:


  • First, choose an option from the drop-down list in order to find any term of geometric Sequence.
  • Now, substitute the corresponding values according to your selection.
  • Click on the calculate button to see the results.


  • Geometric Sequence
  • First Term of the Sequence
  • Common Ratio
  • Nth term and the sum of geometric series


What Are The Main Types of Sequence?

Types of Series and Sequence

  • Arithmetic Sequences
  • Harmonic Sequences
  • Geometric Sequences
  • Fibonacci Numbers


From the source of Wikipedia: Geometric progression, Elementary properties, Derivation, Complex numbers, Product.

From the source of  Purple Math: Find the common difference, Find the common ratio, arithmetic, and geometric sequences, adding (or subtracting) the same values.