First term of the sequence

Common difference

N

N-th term of the sequence

Sum of first N terms

The arithmetic sequence calculator finds the nᵗʰ term and the sum of a sequence with a common difference.

This arithmetic sequence calculator (also called the arithmetic series calculator) is a handy tool for analyzing a sequence of numbers that is created by adding a constant value each time. You can use it to find any property of the sequence - the first term, common difference, nᵗʰ term or sum of the first n terms. You can dive straight into using it or read on to discover how it works. We will explain the sequence equation that the calculator uses and hand you the formula for finding arithmetic series (sum of a sequence).

An arithmetic sequence is a set of numbers. Each consecutive number is created by adding a constant number (called the **common difference**) to the previous one. Such sequence can be finite when it has a determined number of terms (for example, 20) or infinite if we don't specify the number of terms.

Each arithmetic sequence is uniquely defined by two coefficients: the **common difference** and the **first term**. If you know these two values, you are able to write down the whole sequence.

Some examples of an arithmetic sequence include:

- 3, 5, 7, 9, 11, 13, 15, 17, 19, 21...
- 6, 3, 0, -3, -6, -9, -12, -15...
- 50, 50.1, 50.2, 50.3, 50.4, 50.5...

Can you find the common difference of each of these sequences?

Let's assume you want to find the 30ᵗʰ term of any of the aforementioned sequences. Writing down the first 30 terms would be tedious and time consuming. You probably noticed, though, that you don't have to write them all down! It's enough if you add 29 common differences to the first term.

Let's generalize this statement to formulate the arithmetic sequence equation. It is the formula for any nᵗʰ term of the sequence.

`a = a₁ + (n-1)d`

where:

**a**is the nᵗʰ term of the sequence,**d**is the common difference and**a₁**is the first term of the sequence.

Our arithmetic sequence calculator can also find the sum of the sequence (called the arithmetic **series**) for you. You can do it by yourself, too - it's not that hard!

Look at the first example sequence: 3, 5, 7, 9, 11, 13, 15, 17, 19, 21. We could sum all of the terms by hand, but it is not necessary. Let's try to sum the terms in a more organized fashion. We will add the first and last term together, then the second and second-to-last, third and third-to-last etc. You will quickly notice that:

- 3 + 21 = 24
- 5 + 19 = 24
- 7 + 17 = 24

The sum of each pair is constant and equal to 24. That means that we don't have to add all numbers. All you have to do is to add the first and last term of the sequence and multiply that sum by the number of pairs (that means by n/2).

Mathematically,

`Σ = n/2 * (a₁ + a)`

Substituting the equation for nᵗʰ term,

`Σ = n/2 * [a₁ + a₁ + (n-1)d]`

After simplification,

`Σ = n/2 * [2a₁ + (n-1)d]`

This formula will allow you to find the arithmetic series.

Obviously, our arithmetic sequence calculator is not able to analyze any other type of sequence. For example, the sequence 2, 4, 8, 16, 32... does not have a common difference. If you want to analyze it, feel free to check out our geometric sequence calculator.

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