Enter the values in the below input boxes to calculate the nth term and sum of arithmetic progression by using arithmetic sequence/series calculator.
Nth Term Calculator
Arithmetic sequence calculator is an online tool that calculates:
- Arithmetic Sequence
- Nth term value
- Sum of arithmetic sequence
What is an arithmetic sequence?
Arithmetic sequence can be defined as,
“An arithmetic sequence is a sequence where each term increases by adding or subtracting some constant value known as common difference (d).”
Arithmetic sequence is commonly known as arithmetic series and arithmetic progression as well.
Arithmetic sequence formula
Formula to find nth term is:
nth term = a + (n - 1)d
Formula to find sum of arithmetic progression is:
S = n/2 × [2a₁ + (n - 1)d]
Where:
- a refers to nᵗʰ term of the sequence,
- d refers to common difference, and
- a₁ refers to first term of the sequence.
There is no specific formula to find arithmetic sequence. In the next section, we will explain the method to calculate arithmetic sequence using common difference and first term.
Finding nth term, arithmetic sequence and its sum
For the calculation of nth term, arithmetic sequence and its sum, you can simply use the arithmetic series calculator above.
Example:
Find the nth term and sum of arithmetic sequence for 15 number of terms if first term is 5 and difference is 4.
Solution:
Step 1:Identify the values.
n = 15
a = 5
d = 4
Step 2:Use arithmetic sequence formula and place the values.
For finding the nth term
nth term = a + (n - 1)d
= 5 + (15 - 1) × 4
= 61
nth term = 61
For finding the sum of arithmetic sequence
S = n/2 × [2a₁ + (n - 1)d]
S = 15/2 × [2(5) + (15 - 1) × 4]
S = 495
For finding the Arithmetic Sequence
Add common difference in the first term to get the arithmetic sequence. Keep adding the common difference in the preceding number till you get last number in the sequence.
a1 = 5
a2 = a1 + d = 5 + 4 = 9
a3 = a2 + d = 9 + 4 = 13
a4 = a3 + d = 13 + 4 = 17
a5 = a4 + d = 17 + 4 = 21
a6 = a5 + d = 21 + 4 = 25
a7 = a6 + d = 25 + 4 = 29
a8 = a7 + d = 29 + 4 = 33
a9 = a8 + d = 33 + 4 = 37
a10 = a9 + d = 37 + 4 = 41
a11 = a10 + d = 41 + 4 = 45
a12 = a11 + d = 45 + 4 = 49
a13 = a12 + d = 49 + 4 = 53
a14 = a13 + d = 53 + 4 = 57
a15 = a14 + d = 57 + 4 = 61
Arithmetic Sequence = 9, 13, 17, 21, 25, 29, 33, 37, 41, 45, 49, 53, 57, 61
Use the arithmetic progression calculator above to verify the value of nth term and arithmetic sequence.
Use this handy arithmetic sequence calculator to analyze a sequence of numbers you can generate by adding a constant number each time. Instead of performing the calculations manually with the arithmetic sequence formula, you can use the arithmetic series calculator to find a property of the sequence.
- How to use the arithmetic sequence calculator?
- What is the formula of arithmetic sequence?
- How do you write an arithmetic sequence?
- What is the difference between series and sequence?
- What is arithmetic series examples?
How to use the arithmetic sequence calculator?
The arithmetic sequence calculator is a handy tool which makes your life a lot easier. Simply inputting given values in the required fields allows the calculator to do the work for you so you don’t have to put in the effort. Here are the steps to follow for using this arithmetic series calculator:
- First, enter the value of the First Term of the Sequence (a1).
- Then enter the value of the Common Difference (d).
- Finally, enter the value of the Length of the Sequence (n).
- After entering all of these required values, the arithmetic sequence calculator automatically generates for you the values of the n-th Term of the Sequence and the Sum of the First Terms.
What is the formula of arithmetic sequence?
Before answering this question, let’s find out the definition of the term “sequence.” In mathematics, the definition of a sequence is a collection of objects like letters or numbers which appear in a specific order or arrangement. We call these objects terms or elements. It’s common to find the same object appearing several times in a single sequence.
Using this definition, an arithmetic sequence also refers to a collection of objects but in this case, the objects or elements are numbers. In a sequence, you create consecutive numbers as you add a constant number known as the “common difference” to the last one.
You can consider the sequence finite when you set a specific number of elements or infinite if you don’t specify the number of elements. An arithmetic sequence is uniquely defined by the first term and the common difference. As long as you have these values, you can come up with the whole sequence.
Let’s have an example to help you understand the concept better. For instance, you want to find the 32nd term of a sequence. In such a case, writing the first 32 terms would be both time-consuming and tedious. Fortunately, you don’t have to write all these numbers down as long as you have the first term and the common difference.
Using the arithmetic sequence formula, you can solve for the term you’re looking for. This is the formula for any nth term in an arithmetic sequence:
a = a₁ + (n-1)d
where:
a refers to the nᵗʰ term of the sequence
d refers to the common difference
a₁ refers to the first term of the sequence.
You can use this arithmetic sequence formula whether the value of the common difference is zero, negative or positive. Then you can use the arithmetic sequence calculator to check if you performed the calculation correctly.
How do you write an arithmetic sequence?
There are different kinds of arithmetic sequences you can write. You can even use some types of arithmetic sequences in your daily life. As aforementioned, an arithmetic sequence is a sequence of numbers where each successive pair has the same difference.
When you see an arithmetic sequence, you might not think that you can use them for anything important. But these sequences have a lot of practical applications. Since all kinds of arithmetic sequences follow a pattern, it’s quite easy to write them down as long as you know the common difference.
What is the difference between series and sequence?
You can also use this arithmetic sequence calculator as an arithmetic series calculator. But even if you choose to write the sequence down manually, this isn’t that much of a challenge. Let’s have an example of an arithmetic sequence:
3, 5, 7, 9, 11, 13, 15, 17, 19, 21
You can sum all of these terms manually but this isn’t necessary. Instead, you can try to find the sums of the terms in a more organized way. Do this by adding the first term and the last term, the second term and the second-to-the-last term, and so on. Soon, you’ll notice a pattern:
3 + 21 = 24
5 + 19 = 24
7 + 17 = 24
As you can see, the sum of each of the pairs is equal to 24, a constant value. Therefore, you don’t need to add all of the numbers because you’ll just get the same answer. To find the common difference, all you have to do is add the first term and the last term then multiply the sum you get by how many pairs there are.
As an equation, you have:
S = n/2 * (a₁ + a)
Then substitute the equation for the nth term:
S = n/2 * [a₁ + a₁ + (n-1)d]
To make the equation simpler, modify the formula to make it:
S = n/2 * [2a₁ + (n-1)d]
What is arithmetic series examples?
Let’s take a look at some examples of arithmetic series:
50, 50.1, 50.2, 50.3, 50.4, 50.5…
3, 5, 7, 9, 11, 13, 15, 17, 19, 21…
6, 3, 0, -3, -6, -9, -12, -15…
Now that you know how to find the common difference, can you solve for the common differences of these? From these examples, you can see that common differences don’t always have to be natural numbers. They can also be fractions. These values don’t even have to be positive values!
If you have a common difference with a positive value, you call this an increasing arithmetic sequence. Conversely, if the common difference has a negative value, you would call this a decreasing arithmetic sequence. If the common difference is equal to zero, then you have a monotone sequence.
Now look at this sequence:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, …
For this sequence, can you find the common difference? In fact, you can’t because this isn’t an arithmetic sequence. This special sequence is the “Fibonacci sequence” and you can only solve for the next term by taking the sum of the two terms that come before it.