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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)




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Our geometric sequence calculator helps you to find geometric Sequence, first term, common ratio, and the number of terms.

This geometric series calculator provides step-wise calculation 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”.

In other words, a geometric sequence or progression is an item in which another item varies each term by a common ratio. When we multiply a constant (not zero) by the previous item, the next item in the Sequence appears. It is expressed as follows:

$$ a, a(r), a(r)^2, a(r)^3, a(r)^4, a(r)^5and so on. $$

where a is the first item and r is a common ratio.

How to Find Common Ratio?

First term is = a Consider the series is \( a, ar, a(r)^2, a(r)^3, a(r)^4…… \)

Second term is = ar

Similarly, the nth item is, \( k_n = ar^{n-1} \)


Common ratio = (Any item) / (Preceding item)

$$ = k_n / k_{n-1} $$

$$ = (ar^{n – 1} ) /(ar^{n – 2}) = r $$

Thus, the general term of a Geometric progression is given by \( ar^{n-1} \) and the general form of a Geometric sequence is \( a + a(r) + a(r)^2 + a(r)^3 + ….. \)


Find the common ratio, where 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 to Find the Sum of a Geometric Series?

Let’s \( a, ar, a(r)^2, a(r)^3, a(r)^4, ……ar^{n-1} \) 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


Find the sum of first n-terms of geometric sequence, where first term (a_1) = 2, common ration (r) = 2, and sum of first n-terms = 4.


The geometric sequence formula calculator finds the sum of geometric series by:

$$ S_n = a_1 . (1 – r^n) / 1 – r $$

The sum of geometric series sum calculator substitutes the given values in formula:

$$ S_n = a_1 . (1 – r^n) / 1 – r $$

$$ 4 = 2 * (1 – 2^n) / 1 – 2 $$

$$ (1 – 2^n) = 4 / 2 * (1 – 2) $$

$$ (1 – 2^n) = – 2 $$

$$ 2^n = 3 $$

$$ Log (2^n) = log (3) $$

$$ n . log (2) = log (3) $$

$$ n = 1.5849 $$

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 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 Sequence 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}.

How Geometric Sequence Calculator Works?

An online 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.


The geometric sum calculator provides the step-by-step solution and calculates:

  • Geometric Sequence: find the n-th terms, the sum of n terms, sequence of n terms, and display a graph.
  • 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.