# Geometric Sequence Calculator

Find:

First term (a₁)

Common ratio (r)

Number of terms (n)

Enter n-th term a(n)

Sum of first n-terms S(n)

a

=

Table of Content
 1 Geometric Progression or Geometric Sequence: 2 Nth Term of Geometric Sequence: 3 How to Find Common Ratio? 4 How to Find the Sum of a Geometric Series? 5 Infinite Geometric Sequence: 6 Geometric Progression Formulas 7 What are the main types of Sequence? 8 How to know if it is arithmetic or geometric sequence?
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An online 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.  Continue reading to learn how to find common ratios, the sum of finite geometric series, and much more.

## 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.

Note: It should be noted that if we separate the next item from the previous item, then we will get a value corresponding to the common ratio.

Suppose we divide the third term by second term, and we get:

$$A(r)^3/a(r)^2 = r$$

### Geometric Progression or Geometric Sequence:

The general form of Geometric Progression is:

$$a, a(r), a(r)^2, a(r)^3, a(r)^4, …, a(r)^n$$

Where,

an = nth term

a = First term

r = common ratio

However, an online Arithmetic Sequence Calculator allows you to compute the nth value, sum, and Arithmetic sequence.

Example:

Find Geometric Sequence, when the first term is 2, common ratio “r” is 4, and the number of terms is 10.

Solution:

A_1 = 2, r = 4, n = 10

The geometric sequence calculator finds the Nth term of geometric Sequence is:

A_n = a_1 * r{n-1}

A_{10} = 2 * (4)^{10-1}

A_{10} = 2 * 4^9

A_{10} =2 * 262144

A_{10} = 524288

The sum of geometric series calculator find the sum of the first n-terms:

S_n = a_1 * (1 – r^n) / 1 – r

S_{10} = 2 * (1 – 4^{10}) / 1 – 4

S_{10} = 2 * (1 – 1048576) / -3

S_{10} = 699050

The sum of the common ratio calculator determines the first ten terms of the Sequence are:

2, 8, 32, 128, 512, 2048, 8192, 32768, 131072, ….

### Nth Term of Geometric Sequence:

Suppose “r” be the common ratio and “a” be the first item for a Geometric progression.

Then the second item of the series, $$a^2 = a * r = ar$$

Third item, $$a^3 = a^2 * r = a(r) * r = a(r)^2$$

Therefore, nth term, $$a_n = ar^{n-1}$$

Hence, the geometric sequence formula used by the geometric sequence calculator to find the nth term of Geometric series is:

$$A_n = ar^{n-1}$$

### 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, $$k_n = ar^{n-1}$$

Thus,

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 $$a + a(r) + a(r)^2 + a(r)^3 + …..$$

Example:

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

Solution:

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

However, an online Geometric Mean Calculator allows you to compute the geometric mean for a given data set of percentages or numbers.

Example:

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.

Solution:

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:

$$S_n = a_1 . (1 – r^n) / 1 – r$$

$$4 = 2 * (1 – 2^n) / 1 – 2$$

$$(1 – 2^n) = 4 / 2 * (1 – 2)$$

$$(1 – 2^n) = – 2$$

$$2^n = 3$$

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

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

$$n = 1.5849$$

### Infinite Geometric Sequence:

The terms of a finite Geometric progression can be written as $$a, ar, a(r)^2, a(r)^3, a(r)^4, ……ar^{n-1}$$

And the items $$a, ar, a(r)^2, a(r)^3, a(r)^4, ……ar^{n-1}$$, is called finite geometric series.

The sum of finite Geometric series is given by:

$$S_n = a[(r^n-1)/(r-1)] if r ≠ 1$$

Terms of an infinite Geometric Progression can be written as $$a, ar, a(r)^2, a(r)^3, a(r)^4, ……ar^{n-1},…….$$

$$a, ar, a(r)^2, a(r)^3, a(r)^4, ……ar^{n-1},…….$$ is called infinite geometric series.

Example:

Find the sum of the infinite geometric series 64 + 32 + 16 + 8 + 4 + 2

Solution:

First, the infinite geometric series calculator finds the constant ratio between each item and the one that precedes it:

$$R = 32/64$$

$$=1 / 2$$

Now, geometric sequence calculator substitute r=1/2 and a=64 into the formula for the sum of an infinite geometric series:

$$s=64 / (1−1/2) = 64 / (1/2) = 128$$

## 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:

### Input:

• 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.

### Output:

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

## FAQ:

### What are the main types of Sequence?

Types of Series and Sequence

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

### How ​​to know if it is arithmetic or geometric sequence?

If the continuous terms are in a constant ratio, then the Sequence is geometric. On the other hand, if there is a constant difference between consecutive elements, then the Sequence is called an arithmetic sequence.

## Conclusion:

Use this online geometric sequence calculator to evaluate the nth term and the sum of the first n terms of the geometric sequence. It displays complete calculations for finding the sequence, sum of series, and common ratios. This calculator provides all calculations in a fraction of a second using the geometric sequence formula.

## Reference:

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

From the source of Lumen Learning: Definition of geometric progression, Behavior, Summing the First n Terms in a Geometric Sequence, Infinite Geometric Series, Applications of Geometric Series, Repeating Decimal, Archimedes’ Quadrature of the Parabola, Fractal Geometry.

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