Arithmetic Series Calculator

This arithmetic series calculator works out the nth term and the total sum of an arithmetic sequence, a list of numbers where each term increases or decreases by the same fixed amount, known as the common difference. You enter three values: the first term, the common difference, and the number of terms to add together. The calculator instantly returns the sum of the whole series as the headline result, alongside the nth term, the final value reached, and a confirmation of how many terms were included. Behind the scenes it applies the standard formula, S equals n over 2 times (2 times the first term plus n minus 1 times the common difference), the same shortcut a young Gauss famously used when he paired the first and last terms, then the second and second last, and so on, to add a long run of numbers in seconds rather than one at a time. Arithmetic series turn up whenever amounts step by a constant rate, such as stacking objects in rows, scheduling repeating tasks, or savings contributions that rise by a fixed sum each period. Change any input and the results update immediately, making this a quick way to check homework, test starting points, or see how the common difference changes the total. Results are rounded for display, so treat them as a maths aid rather than a certified answer.

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First term a1

Common difference d

Number of terms n

nth term

Number of terms

The formula

The nth term is a_n = a1 + (n minus 1) d. The sum of the first n terms is S = n/2 times (2 a1 + (n minus 1) d), which also equals n times the average of the first and last terms.

Worked example

For first term 1, common difference 1, ten terms (1 + 2 + ... + 10), the tenth term is 10 and the sum is 10/2 times (2 + 9) = 55. Enter 1, 1, 10 to confirm.

Frequently asked questions

What is an arithmetic series?

The sum of an arithmetic sequence, where consecutive terms differ by a constant common difference.

What is the sum formula?

S = n/2 times (first term plus last term), or equivalently n/2 times (2 a1 + (n minus 1) d).

What is the nth term?

a1 + (n minus 1) d, the first term plus n minus 1 steps of the common difference.

Who this calculator is for

This calculator is for algebra students and anyone needing a quick, reliable result.

What this calculator assumes

You enter valid numbers.
The standard method is applied.
Results are rounded for display.

Formula and sources

Related calculators

Quadratic Formula Calculator: solve a quadratic.
Scientific Calculator: general maths.
Matrix Calculator: matrix operations.

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