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Geometric Sequence Calculator + Online Solver With Free Easy Steps

The Geometric Sequence Calculator allows you to calculate the common ratio between a sequence of numbers.

The Geometric Sequence Calculator is a powerful tool that has various applications. An essential application of the Geometric Sequence Calculator is finding progressing interest in a saving account. Other powerful applications can be found in biology and physics.

What Is a Geometric Sequence Calculator?

A Geometric Sequence Calculator is an online tool used to calculate the common ratio between a number sequence.

The Geometric Sequence Calculator requires four types of input: the $j^{th}$ term (Xj) , the $k^{th}$ term (Xk), the position of (Xj) term, and the position of (Xk) term. The Geometric Sequence Calculator then computes the common ratio between this sequence and provides the results.

How To Use The Geometric Sequence Calculator?

You can use the Geometric Sequence Calculator by entering the mathematical values into their respective fields and clicking the “Submit” button. The Geometric Sequence Calculator then provides the results.

The step-by-step instructions for using a Geometric Sequence Calculator can be found below.

Step 1

First, you will need to add the $j^{th}$ term into your calculator.

Step 2

After adding the $j^{th}$ term, you will then add the position where the $j^{th}$ term is located. 

Step 3

After entering the $j^{th}$ term and its position, the value of the  $k^{th}$ term is added into its respective box.

Step 4

Similar to step 2, enter the position of the $k^{th}$ term.

Step 5

Finally, after plugging in all the values, click the “Submit” button. The Geometric Sequence Calculator displays the common ratio and equation used in a separate window.

How Does a Geometric Sequence Calculator Work?

The Geometric Sequence Calculator works by using the $k^{th}$ and $j^{th}$ terms along with their positions to find the common ratio between each number in the sequence. The common ratio is shown in a separate window along with the equation used to derive the ratio. The equation used is as follows:

\[ r = \frac {X_{n}}{X_{n-1}} \]

In order to fully grasp the concept behind this calculator, let us first look at some important concepts related to the workings of the calculator.

What Is a Geometric Sequence?

A geometric sequence is a sequence in which all but the first number are derived by multiplying the preceding one by a constant, non-zero amount referred to as the common ratio. The following formula is used to derive the common ratio.

\[ a_{n} = a_{1}r^{n-1} \]

We will discuss the derivation of this equation in a while.

First, it is essential to realize that despite the geometric sequences’ constant multiplication of the numbers, it is different from factorials. However, they have similarities, such as the relationship of numbers for their GCM (Greatest Common Factor) and LCM (Lowest Common Factor).

This means that the GCF is the smallest value in the sequence. In contrast, the LCM represents the highest value in the series.

What Is Geometric Progression?

A geometric progression is a group of numbers connected by a common ratio, as mentioned previously. The common ratio is the defining function responsible for connecting these numbers in a sequence.

The initial number of the sequence and the common ratio are used to derive recursive and explicit formulas.

Now let us construct an equation we can use to describe geometric progression. For example, let us set the initial term to 1, and the common ratio is set to 2. This means that the first term would be a1 = 1. By using the definition above, we can derive the common ratio equation as a2 = a2 x 2 = 2.

Hence the n-th term of the geometric progression would as the following equation:

\[ a_{n} = 1 \ * \ 2^{n-1} \]

$n$ is the position of the term in the sequence.

Typically, a geometric sequence is written down by starting from the initial number and continuing in ascending order. This helps you calculate the series much more effortlessly.

There are several ways to represent information in mathematics. Similarly, we will be looking at recursive and explicit formulas used to find geometric sequences.

Types of Geometric Progression

Geometric progression has two types that are based on the number of items a geometric progression: Finite geometric progression and Infinite geometric progression. We will discuss both of these types below.

What Is Finite Geometric Progression?

A finite geometric progression is a geometric progression in which terms are written as $a, ar, ar^{2}, ar^{3}, ar^{4},… $. The sum of the finite geometric progressions is found using the equation below.

\[ S_{n} = a[ \frac {(r^{n}-1)}{(r-1)} ] \]

What Is Infinite Geometric Progression?

An infinite geometric progression is a geometric progression in which terms are defined by $a, ar, ar^{2}, ar^{3}, ar^{4},… $. The sum of the infinite geometric progressions can be found using the equation below.

\[ \sum_{k=0}^{\infty} (\frac{a}{r^{k}}) = a(\frac{1}{1-r}) \]

Properties of Geometric Sequence

Here are some properties of Geometric sequence:

  • A new series produces a geometric progression with the same common ratio when each term of a geometric progression is multiplied or divided by the same non-zero quantity. 
  • The reciprocals of the terms also form a geometric progression in a geometric sequence. In a finite geometric progression, the product of the first and last terms is always equal to the product of the terms equally spaced apart from the start and end. 
  • There can be geometric progression if three non-zero quantities a,b,c are equal to $ b^{2} = ac $.
  • The new series also has a geometric progression when the terms of an existing series are chosen at regular intervals.
  • When there are non-zero, non-negative terms in a geometric progression, the logarithm of each term creates an arithmetic progression and vice versa.

Explicit Formula Used In Geometric Sequence

Explicit Formulas are used to define information in the geometric sequence. Derivation of the explicit formula is shown above. We can substitute values and simplify the formula even more to create a general equation.

We substitute the first term with a1 and the ratio with r. The following formula is derived.

\[ a_{n} = a_{1} \ * \ r^{n-1} \]

where,

\[n \in \mathbb{N} \]

Where $ n \in N $ means n = 1,2,3,4,5,….

Now let us look into the recursive formula for a geometric sequence.

Recursive Formula Used In Geometric Sequence

The recursive formula is another way to represent information in a geometric sequence. There are two main parts of a recursive formula. Both of these parts convey different information about the geometric sequences. 

The first part explains how to calculate the common ratio between the numbers. The second part describes the first term in the geometric sequence. We can calculate the common ratio by combining these two pieces of information.

The following equation is the recursive formula:

\[ a_{n} = a_{n-1} \ * \ r \]

ai = x 

Here, the x represents any explicit number that can be used. The equation is similar to the explicit formula we looked at previously.

What Is a Common Ratio In Geometric Sequence?

A common ratio is a number multiplied or divided at intervals between numbers in a geometric sequence. This is a common ratio because the answer would always be the same if you divided two successive digits. It does not matter where you select the terms — they need to be next to each other. 

Generally, we represent the general progression as a1, (a1r), (a2r), (a3r),…  here a1 is the first term, (a1r) is the second term, and so on. The common ratio is denoted by r. 

Looking at the above representation of general progression, we can derive the following equation for the common ratio.

\[ r = \frac {a_{n}}{a_{n-1}} \]

Arithmetic Sequences and Geometric Sequences

An arithmetic sequence is a sequence in which the difference between two consecutive numbers is the same. It simply means that the last number in the series is multiplied by a predetermined integer to determine the following number. 

Here is an example of how arithmetic sequences are represented:

 a, a+d, a + 2d , a + 3d, a + 4d,… 

Here $a$ is the first term, and d is the common difference between the terms.

In contrast, geometric sequences are numbers that have a common ratio between each value. The common ratio is the same for each consecutive value. The following number in the sequence is calculated by multiplying the common ratio with the term.

Here is an example of how geometric sequences can be represented:

\[ a, ar, ar^{2}, ar^{3}, ar^{3},… \]

Here, a is the first term and r is the common ratio between the sequences.

The following table describes the difference between geometric and arithmetic sequences.

Arithmetic SequenceGeometric Sequence 
A series of numbers known as an arithmetic sequence varies from one another by a predetermined amount with each successive number. A series of integers is a geometric sequence if each succeeding element is produced by multiplying the previous value by a fixed factor. 
A common difference exists between succeeding numbers. A common ratio exists between consecutive numbers. 
Arithmetic operations like addition and subtraction are used to get the following values. Represented by d.Multiplication and division are used to calculate the consecutive numbers. Represented by r.

Example:

 5, 10 , 15 , 20,… 

Example:

 2, 4, 8 , 16 ,… 

 

How Are Geometric Sequences Used in Real Life?

Geometric sequences are widely used in several applications, and one common real life application of geometric sequences is in calculating interest rates. 

When calculating a term in a series, mathematicians multiply the sequence’s starting value by the rate increased to a power of one below the term number. A borrower can determine from the sequence how much his bank anticipates him to repay using simple interest.

Geometric sequences are also used in fractal geometry while calculating a self-similar figure’s perimeter, area, or volume. For example, the area of the Koch snowflake can be calculated by the union of infinitely placed equilateral triangles. Each small triangle is $ \frac {1}{3} $ of that of the larger triangle. The following geometric sequence is generated.

\[ 1 + 3( \frac{1}{9}) + 12(\frac{1}{9})^{2} + 48(\frac{1}{9})^{3} +… \]

Biologists also use a geometric sequence. They can calculate the population growth of bacteria in a petri dish using geometric sequences. Marine biologists can also use geometric sequences to approximate the population growth of fish in a pond by using geometric sequences.

Physicists also use geometric sequences in calculated the half-life of a radioactive isotope. Geometric sequences are also used in several physics experiments and equations.

A geometric sequence is a very versatile mathematical law that is used in various fields around the world. 

History of Geometric Sequence Calculators

Geometric sequences were first used 2,500 years ago by Greek mathematicians. The mathematicians felt that walking from place to place was a tiresome task. Zeno of Elea pointed out a paradox, suggesting that one must travel half the distance to reach a destination.

Once going half the distance, he would have to travel half the space again. This paradox would continue until infinity was reached. However, this paradox was considered wrong later on.

In 300 B.C Euclid of Alexandria wrote his book “The Elements of Geometry.” The book contained the first interpretation of geometric sequences. The text was later deciphered, and Euclid’s equations for geometric sequences were extracted. Different mathematicians further simplified these equations.

In 287 B.C, Archimedes of Syracuse used geometric sequences to calculate the area of a parabola enclosed in straight lines. Archimedes’ implementation of geometric sequences allowed him to dissect the area in an infinite number of triangles. The area of a parabola can easily be computed using integration today.

In 1323, Nicole Oresme proved that the series $ \frac{1}{2} + \frac{2}{4} + \frac{3}{8} + .., $ consolidates to 2. Nicole derived this proof using geometric sequences.

Geometric sequences have been used throughout history and have proven to be significant in deriving new proofs. We have discussed the importance and derivation of geometric sequences throughout the years.

Solved Examples

The Geometric Sequence Calculator can easily calculate the common ratio between two consecutive numbers. Here are some solved examples that use the Geometric Sequence Calculator.

Example 1

A high school student is presented with a geometric sequence of 2, 6, 18, 54, 162,…. He is required to find the common ratio $r$. Calculate the common ratio using the geometric sequence provided.

Solution

To solve this problem, we can use the Geometric Sequence Calculator. First, we select any two consecutive values from the geometric sequence provided. We select the values 6 and 18. The positions of these terms are 1 and 2.

Enter the numbers from the geometric sequence into the Xk and Xj boxes, then add the position of each term into their respective boxes.

Click the “Submit” button and you will be presented with the common ratio. The results can be seen below:

Input:

\[ \sqrt[2-1]{\frac{18}{16}} \]

Exact result:

3

Number name:

three 

Example 2

While experimenting, a physicist stumbles upon a geometric sequence of  3840, 960, 240, 60, 15,… . To complete his experiment, the physicist derives a ratio common for numbers in a geometric sequence. Using the Geometric Sequence Calculator, find this ratio.

Solution

Solving this problem requires us to use The Geometric Sequence Calculator. First, we need to select two numbers next to each other from the geometric sequence provided. Suppose we select the numbers 960 and 240. We then note down the positions of the terms, which are 2 and 3, respectively. 

We then enter our selected numbers and add them to the Xk and Xj boxes. After adding the numbers, we input the positions of the terms. Finally, after all these steps, we click the “Submit” button and our ratio is shown in a new window.

The results are shown below:

Input:

\[ \sqrt[3-2]{\frac{240}{960}} \]

Exact result:

\[ \frac{1}{4} \]

Example 3

A college student is given an assignment where he has to find the common ratio of the following geometric sequence.

10,20,30,40,50,… 

Using The Geometric Sequence Calculator, find the common ratio of the sequence.

Solution

We will use the Geometric Sequence Calculator to solve this problem. First, we select two numbers from the sequence. We choose 30 and 40, keeping in mind that the numbers should be consecutive. We also need to know the positions of these terms, which are 3 and 4.

After gathering all the data from the geometric sequence, we first plug in the number pairs in the Xk and Xj boxes. We then add the position of the terms in their respective boxes. To find the result, we click the “Submit” button. A new window displaying the results is opened on our Geometric Sequence Calculator. You can look at the results below.

Input:

\[ \sqrt[4-3]{\frac{40}{30}} \]

Exact result:

\[ \frac{1}{4} \]

Example 4

A biology student is experimenting with a specific type of bacteria. The student looks at the growing population of bacteria in a petri dish and generates a geometric sequence of 2,4,16, 32, 64,…. Find the common ratio using the geometric sequence provided.

Solution

Using our Geometric Sequence Calculator, we can easily find the common ratio of the geometric sequence. First, we select a pair of numbers that are consecutive to each other. In this example, we select 32 and 64. After selecting the pair, we figure out their positions, which are 4 and 5. 

Once we have gathered the necessary information, we can start inputting values into the Geometric Sequence Calculator. First, we add the pair numbers in the Xk and Xj boxes, then we add the position of terms in their respective boxes. Finally, we click the “Submit” button, which displays the results in a new window. The results can be seen below.

Input:

\[ \sqrt[5-4]{\frac{64}{32}} \]

Exact result:

 2 

Number name

 two 

Example 5

During his research, a mathematics professor came across a geometric sequence 4, 20, 100, 500,…. The professor wants to find a common ratio that can relate to the entire sequence. Calculate the common ratio of the geometric sequence given above.

Solution

Using our reliable Geometric Sequence Calculator, we can easily solve this problem. First, we select two numbers from the geometric sequence; these numbers should be consecutive. We pick 20 and 100. After selecting these values, we find the positions of these terms, which are 2 and 3.

Now we open the first two numbers into the  Xk and Xj boxes. Subsequently, we add the positions of the terms in their respective boxes. After inputting all the necessary data into our Geometric Sequence Calculator, we hit the “Submit” button. A new window will appear, showing the results from the calculator. The results are shown below.

Input:

\[ \sqrt[2-3]{\frac{100}{20}} \]

Exact result:

Number name:

 five

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