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An arithmetic sequence calculator is a convenient tool for evaluating a sequence of numbers, which is generated each time by adding a constant value. It is also known as the arithmetic series calculator. Any property of the sequence can be calculated, such as common difference, nth term, the sum of the first n terms, or the first term. If you want to see how this recursive sequence calculator calculates the arithmetic sequence, read this complete post.
In this post, we will discuss the arithmetic sequence, its formula, examples, and many other things to clear the concept of the arithmetic sequence.
How to use Arithmetic Sequence Calculator?
Our sequence finder is the best tool in the market to find a specific arithmetic sequence term or sum of a series. To use this sequence calculator, follow the below steps.
- The first input box is divided into two categories. Enter the location of any term and its value in this input box. For example, if you have a sequence of 3, 5, 7, 9, the first term will be 3. It is the 1st term in the sequence. You should enter 1 on the left side of the first input box and 3 on the right side. Please refer to the image below.
- Enter the common difference in the next input box and then enter the total number of terms in the corresponding input box.
- After entering the values, press the Calculate You will get the detailed results as soon as you hit that button.
It will give you the complete table depicting each term in the sequence and how it is evaluated. You can also find the graphical representation of the sequence, which makes it very easy to understand. Moreover, you will get all the steps that are performed by this calculator to calculate the terms in the sequence and sum of the sequence. It could come in very handy to prepare for exams or your next assignment.
What is an arithmetic sequence?
You must first figure out what the word sequence means to understand the concept of the arithmetic sequence. A set of objects, including numbers or letters in a certain order, is known as a sequence in mathematics. The sequence objects are known as terms or elements. It is quite normal to see the same object in one sequence many times.
Arithmetic sequence definition can be interpreted as: a set of objects that comprises of numbers is an arithmetic sequence. A constant number known as the common difference is applied to the previous number to create each successive number. An arithmetic sequence of this kind can be finite when the number of terms is defined, or infinite when the number of terms is not defined.
There are two coefficients for each arithmetic sequence: the first term and the common difference. You will be able to write down the entire sequence if you know these two values.
As soon as you plunge into the topic of arithmetical sequence, you'll probably find uncertainty about some terms. It occurs due to the use of various naming conventions.
The arithmetic sequence and series are two of the most common terms you may come across. An arithmetic sequence is also referred to as arithmetic progression, while the arithmetic series is known as a partial sum.
Arithmetic sequence formula
It will be time consuming and boring task to write down all the terms if you want to find a specific term in a sequence. However, you probably noticed that all of them must not be written down. If you want to find the 20th term in a sequence, you may add 19 common differences to the first term of the sequence.
We can use the arithmetic sequence formula to find any term in the sequence. Arithmetic sequence equation can be written as:
\(a_n = a_1 + (n-1)d\)
In this equation:
\(a\) refers to the \(n^{th}\) term of the sequence,
\(a_1\) refers to the first term of the sequence,
\(d\) refers to the common difference and
\(n\) refers to the length of the sequence.
The above formula is an explicit formula for an arithmetic sequence. All terms are equal to each other if there is no common difference in the successive terms of a sequence. In this case, there would be no need for any calculations.
Arithmetic sequence examples
The arithmetic sequence can be written as:
\(2, 4, 6, 8, 10, 12, 14, 16, 18...\)
\(1.3, 1.5, 1.7, 1.9, 1.11, 1.13, 1.15...\)
\(5, 2, -1, -4, -7, -10, -13, -16...\)
If you can calculate the common difference in these sequences, you will be able to find any term in the sequence. By subtracting a term from the previous term in an arithmetic sequence, you can find the common difference d.
In this sequence: \(2, 4, 6, 8, 10, 12, 14, 16, 18...\)
Common difference is 2, because \(4 - 2 = 2\)
You can see that the common difference does not have to be a natural number based on these examples of arithmetic sequences above. It could be a decimal or fraction. It doesn't have to be a +ve number.
Let's find the 10th term in the above sequence by using the arithmetic sequence formula. As we know, n refers to the length of the sequence, and we are about to find the 10th term in the sequence, which means the length of the sequence will be 10. Follow these steps to find a specific term in an arithmetic sequence.
Step 1: Write down the sequence.
\(2, 4, 6, 8, 10, 12, 14, 16, 18...\)
Step 2: Find the common difference d.
\(d = 2\)
Step 3: Write down the formula of the arithmetic sequence.
\(a_n = a_1 + (n-1)d\)
Step 4: Substitute the values in the equation.
As we know,
\(a_1= 2, n = 10,\) and \(d = 2\)
\(a_{10} = 2 + (10 - 1) 2 = 20\)
So the 10th term of this arithmetic sequence would be 20. You can also use our above arithmetic sequence formula calculator to make this whole process convenient.
What is the difference between a series and a sequence?
By using our arithmetic calculator, you can also find the sum of the sequence, also known as the arithmetic series. You can also do it by yourself, but using our sum of arithmetic sequence calculator above, you can save a lot of time.
Let's use the sequence to find the sum of the series.
\(2, 4, 6, 8, 10, 12, 14, 16, 18...\)
All term in an arithmetic sequence can be added by hand, but there is no need to do it. Let's try to add them in a more ordered way. We must add together the first term in the sequence and final one, the second term in the sequence and the second last one, and so on. In this way, it will be much easier to calculate.
\(2 + 18 = 20\)
\(4 + 16 = 20\)
\(6 + 14 = 20\)
Every time we add two terms in this way, we will get a constant answer. That means not all numbers need to be added. You simply have to add the first and last sequence term and then multiply the answer by n/2.
Arithmetic series equation can be written as:
\(S = [2a_1 + (n - 1) d] \times \dfrac{n}{2}\)
Place the required values in the above equation:
\(S = [2 + 2 + (9 - 1) 2] \times \dfrac{9}{2}\)
\(S = 20 \times \dfrac{9}{2}= 90\)
It can also be calculated by using the above recursive formula calculator.