With a geometric sequence calculator, you can calculate everything and anything about geometric progressions. It’s a simple online calculator which provides immediate and accurate results. Let’s cover in detail how to use the geometric series calculator, how to calculate manually using the geometric sequence equation, and more.
How to use the geometric series calculator?
The geometric series calculator or sum of geometric series calculator is a simple online tool that’s easy to use. With it, you can get the results you need without having to perform calculations manually. You can also use the calculator to check the correctness of your answer. Here are the steps in using this geometric sum calculator:
- First, enter the value of the First Term of the Sequence (a1).
- Then enter the value of the Common Ratio (r).
- Finally, enter the value of the Length of the Sequence (n).
- After entering all of the required values, the geometric sequence solver automatically generates the values you need namely the n-th term of the sequence, the sum of the first n terms, and the infinite sum.
What is the geometric sequence equation?
If you want to perform the geometric sequence manually without using the geometric sequence calculator or the geometric series calculator, do this using the geometric sequence equation. Although there is a basic equation to use, you can enhance your efficiency by playing around with the equation a bit.
To modify the equation and make it more efficient, let’s use the mathematical symbol of summation which is ∑. This means that every term after the symbol gets summed up. Therefore, the equation becomes:
S = ∑ P∞
This is the first geometric sequence equation to use and as you can see, it’s extremely simple. However, most mathematicians won’t write the equation this way. Mathematically, geometric sequences and series are generally denoted using the term a∞. Therefore, the equation looks like this:
S = ∑ a∞ = a₁ + a₂ + a₃ + … + a∞
However, this equation poses the issue of actually having to calculate the value of the geometric series. This is why a lot of people choose to use a sum of geometric series calculator rather than perform the calculations manually. Still, understanding the equations behind the online tool makes it easier for you.
Here’s a trick you can employ which involves modifying the equation a bit so you can solve for the geometric series equation:
S = ∑ a∞ = ∑ a₁rⁿ⁻¹ = a₁ + a₁r + a₁r² + … + a₁rᵐ⁻¹
Now you have to multiply both od the sides by (1-r):
S * (1-r) = (1-r) * (a₁ + a₁r + a₁r² + … + a₁rᵐ⁻¹)
S * (1-r) = a₁ + a₁r + … + a₁rᵐ⁻¹ – a₁r – a₁r² – … – a₁rᵐ = a₁ – a₁rᵐ
S = ∑ a∞ = a₁ – a₁rᵐ / (1-r)
The final result makes it easier for you to compute manually. Then you can check if you calculated correctly using the geometric sum calculator.
What is the common ratio of the following geometric sequence?
In layman’s terms, a geometric sequence refers to a collection of distinct numbers related by a common ratio. The common ratio refers to a defining feature of any given sequence along with its initial term. To help you understand this better, let’s come up with a simple geometric sequence using concrete values.
To simplify things, let’s use 1 as the initial term of the geometric sequence and 2 for the ratio. In such a case, the first term is a₁ = 1, the second term is a₂ = a₁ * 2 = 2, the third term is a₃ = a₂ * 2 = 4, and so on. Here, the nth term of the geometric progression becomes:
a∞ = 1 * 2ⁿ⁻¹
n refers to the position of the given term in the geometric sequence
One of the most common ways to write a geometric progression is to write the first terms down explicitly. Then you can calculate any other number in the sequence.
What is sum of geometric series?
The sum of geometric series refers to the total of a given geometric sequence up to a specific point and you can calculate this using the geometric sequence solver or the geometric series calculator. A geometric sequence refers to a sequence wherein each of the numbers is the previous number multiplied by a constant value or the common ratio.
Let’s have an example to illustrate this more clearly. For instance, you’re growing root crops. Let’s assume that for each root crop you plant, you get 20 root crops during the time of harvest. Then when you plant each of those 20 root crops, you get 20 more new ones from each of them.
Therefore, you will have 20 * 20 root crops or a total of 400. If you plant these root crops again, you will get 400 * 20 root crops giving you 8,000! For this example, the geometric sequence progresses as 1, 20, 400, 8000, and so on. As you can see, you multiply each number by a constant value which, in this case, is 20.
This is a real-life application of the geometric sequence. So if you’re a farmer or you’re faced with a similar situation, you can either use the geometric series calculator or perform the calculation manually.
What makes a sequence geometric?
In mathematics, the simplest types of sequences you can work with are the geometric and arithmetic sequences. An arithmetic sequence simply progresses from one term to the next either by subtracting or adding a constant value. The number subtracted or added in an arithmetic sequence is the “common difference.”
A geometric sequence differs from an arithmetic sequence because it progresses from one term to the next by either dividing or multiplying a constant value. Here, the number which you divide or multiply for the progression of the sequence is the “common ratio.” Either way, the sequence progresses from one number to another up to a certain point.