Arithmetic Sequence Calculator

The arithmetic sequence calculator instantly calculates the arithmetic sequence along with the nth term, sum, and number of series. You can instantly figure out the common difference in an arithmetic sequence with the calculator.

What Is Arithmetic Sequence?

In mathematics, it is defined as:

An arithmetic sequence is a list of numbers in which the difference between each successive term remains constant. 

Generally, arithmetic sequence is also known as arithmetic series and arithmetic progression. This sequence can be written in its general form as:

an = a1 + f × (n-1)

Arithmetic Sequence Formula:

The common difference in the specific set of numbers in which each number is the resultant of the sum of previous numbers can either be positive or negative. The sign determines the direction of the sequence. 

• A positive common difference results in a sequence that tends towards positive infinity.
• A negative common difference results in a sequence that tends towards negative infinity.

The arithmetic series formulas are as follows:

For nth Term:

$$ n^{th} Term = a + \left(n-a\right) * d $$

For Sum of Arithmetic Progression:

$$ S = \frac{n}{2} * [2a_{1} +  \left(n-a\right) * d $$

Where;

• a = nᵗʰ term of sequence
• d = Common Difference
• a_1 = First Term 

How To Calculate Arithmetic Sequence?

Let us resolve a couple of examples in complete steps that will help you calculate arithmetic sequences manually! 

Example # 01:

Find the 32nd term of the following arithmetic sequence:

$$ 39, 35, 31, 27, 23, … $$

Solution:

As we have:

$$  a_{1} = 39 $$

$$ d = 35 - 39 = -4 $$$$ n = 32 $$

Now we have 

$$ n^{th} Term = a + \left(n-a\right) * d $$

$$ a_{32} = 39 + \left(31-1\right) * -4 $$

$$ a_{32} = 39 + 31 * -4 $$ 

$$ a_{32} = 39 + 124 $$

$$ a_{32} = 163 $$

Example # 02:

Calculate the sum of up to 10 terms of the arithmetic sequence with the following attributes:

$$ a_{1} = 3 $$ 

$$ d = 2 $$ 

Solution:

Finding nth Term:

$$ n^{th} Term = a + \left(n-1\right) * d $$

$$ n^{th} Term = 3 + \left(10-1\right) * 2 $$

$$ n^{th} Term = 3 + \left(9\right) * 2 $$

$$ n^{th} Term = 3+18 $$

$$ n^{th} Term = 21 $$

Finding Sum up to 10 Terms:

$$ S = \frac{n}{2} * [2a_{1} +  \left(n-1\right) * d] $$

$$ S = \frac{10}{2} * [2 * 3 +  \left(10-1\right) * 2] $$

$$ S = \frac{10}{2} * [6 +  9 * 2] $$

$$ S = 5 * 6 +  9 * 2 $$

$$ S = 30 + 18 $$

$$ S = 48 $$

Writing Arithmetic Series:

Arithmetic Series = 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 + 21

How To Use This Calculator?

The calculator is an extremely fast result-oriented tool. Keep scrolling to know how to use it!

Required Entries:

• Enter the first term (a)
• Put common difference (d)
• Enter nth term number (n)

Result Summary:

• Arithmetic sequence
• Nth term
• Sum from first to nth term
• Complete stepwise calculation

Additional Queries:

What Is The Difference Between Arithmetic Sequence and Series?

An arithmetic sequence is simply the set of objects created by adding a constant value each time. On the other hand, the arithmetic series is the sum of n objects in sequence. 

How Do You Know If a Sequence Is Arithmetic or Geometric?

If it comes about the arithmetic sequence, then it is obtained by maintaining a constant difference between successive numbers and can be instantly determined by a common difference calculator. On the other hand, a geometric sequence has a constant ratio between numbers.

How Do I Find The Common Difference In an Arithmetic Sequence?

Common differences in an arithmetic sequence can be easily determined by the arithmetic sequence calculator because it shows stepwise calculations for arithmetic progression that are obtained by adding a constant number. However, when it comes to manually, you can get the common difference by finding the difference between any two terms in an arithmetic sequence. 

References:

From the source of Wikipedia: Arithmetic progression, Product, Standard deviation