What is an arithmetic sequence?

An arithmetic sequence (also called an arithmetic progression) is a list of numbers where the difference between any two consecutive terms is constant. This constant difference is called d. For example, 3, 8, 13, 18, 23 is an arithmetic sequence with first term 3 and common difference 5. The nth term is given by a plus (n minus 1) times d.

How do you find the sum of an arithmetic sequence?

The sum of the first n terms of an arithmetic sequence is S(n) = n divided by 2 times (2a plus (n minus 1) times d), which can also be written as n divided by 2 times (first term plus last term). This formula was famously discovered by Gauss as a child, who noticed that pairing the first and last terms of a sequence gives the same total as pairing the second and second-last terms, and so on.

What is the difference between an arithmetic and geometric sequence?

In an arithmetic sequence, you add the same constant d to each term to get the next one, so the terms increase (or decrease) at a steady linear rate. In a geometric sequence, you multiply each term by the same constant ratio r, so the terms grow (or shrink) exponentially. A sequence of savings account balances growing by a fixed dollar amount each year is arithmetic; one growing by a fixed percentage is geometric.

Arithmetic Sequence Calculator

An arithmetic sequence is a list of numbers where each term is obtained by adding the same fixed value, called the common difference d, to the previous term. If the first term is a and the common difference is d, the sequence is a, a+d, a+2d, a+3d, and so on. The nth term is found by the formula a(n) = a + (n-1) times d, which makes it easy to jump to any position in the sequence without listing all the intermediate terms. The sum of the first n terms, known as the nth partial sum, is given by S(n) = n divided by 2 times (2a plus (n-1) times d), a formula that works because the sequence is symmetric: pairing the first and last terms, the second and second-last terms, and so on always gives the same total. Arithmetic sequences arise naturally in many contexts: equal monthly repayments on a loan, a person who saves the same extra amount each year, seating rows in a theatre where each row has one more seat than the one before, and depreciation calculated by reducing an asset by the same dollar amount each year. You enter the first term a, the common difference d, and the number of terms n, and the calculator returns the nth term, the partial sum, the mean value of the sequence, and a full list of all n terms. The first term and common difference can be any signed decimal; n must be a positive integer. For very large n, only the result values are displayed, not the full term list.

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First term (a)

Common difference (d)

Number of terms (n)

nth term a(n)

Partial sum S(n)93

Mean of sequence15.50

Last term28

3, 8, 13, 18, 23, 28

Terms list shown for n up to 50. Formula: a(n) = a + (n-1)d. Sum: S(n) = n/2 x (2a + (n-1)d).

How it works

The nth term is calculated as a(n) = a + (n minus 1) times d. The partial sum uses the formula S(n) = n divided by 2 times (first term plus last term) = n divided by 2 times (2a plus (n minus 1) times d). The mean of the sequence equals S(n) divided by n, which for a symmetric arithmetic sequence is simply the average of the first and last terms: (a plus a(n)) divided by 2. The full term list is generated iteratively by starting at a and repeatedly adding d, giving the n terms a, a+d, a+2d, up to a+(n-1)d.

Worked example

Using the defaults a=3, d=5, n=6: the terms are 3, 8, 13, 18, 23, 28. The nth term is 3 + (6-1) times 5 = 3 + 25 = 28. The partial sum is 6 divided by 2 times (2 times 3 + 5 times 5) = 3 times (6 + 25) = 3 times 31 = 93. The mean is 93 divided by 6 = 15.50, which also equals (3 + 28) divided by 2 = 15.50.

Related calculators

Geometric Sequence Calculator: nth term and sum of a geometric (constant-ratio) progression.
Summation Calculator: evaluate any sigma expression with custom upper and lower bounds.
Fibonacci Calculator: the nth Fibonacci number and the full sequence.
Percent Error Calculator: verify how close a manual sequence calculation is to the correct answer.

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