Arithmetic sequence calculator

An arithmetic sequence calculator works with sequences that go up or down by the same amount each time, such as 7, 11, 15, 19. Tell it where the sequence starts, a, and how much it changes by each time, d, and our Sequences and Series Calculator finds the nth term rule, any term you ask for and the sum of the first n terms, all exactly. Or type the first few terms and it finds the rule for you.

The calculator handles arithmetic and geometric sequences, so choose Arithmetic to switch it to this task. It opens on a geometric sequence.

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Sequences and Series Calculator

Find nth terms and sums of arithmetic and geometric sequences

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How to use the arithmetic sequence calculator

1. Choose Arithmetic.
2. Type the First term, a, the Common difference, d and the Term number, n you want. Fractions and negatives both work.
3. Read the answer: the nth term, then the rule uₙ, the sum of n terms and a note on the sum to infinity.
4. Follow the working, which puts your numbers into each formula.

If you only have the terms, choose Find rule and type them with commas, such as 5,11,17,23,29. The calculator finds the differences, says it is arithmetic, and gives the rule and the next terms. It also recognises geometric, quadratic and Fibonacci-type sequences. The graph plots each term against n with the running total. For practice, Practice and help sets random sequence questions with worked solutions.

Arithmetic sequence formulas

Here a is the first term, d is the common difference, n is the term number and l is the last term. The US often writes aₙ for the nth term instead of uₙ.

Worked examples you can check in the calculator

Example 1: the nth term and the sum

For the sequence 7, 11, 15, 19, …, find the 20th term and the sum of the first 20 terms. Here a = 7 and d = 4.

The rule simplifies to uₙ = 4n + 3. Type a = 7, d = 4 and n = 20: the calculator gives u₂₀ = 83, the rule 4n + 3 and S₂₀ = 900.

Example 2: a decreasing sequence

A sequence starts at 50 and goes down by 3 each time. Here d = −3, so u₁₅ = 50 + 14 × (−3) = 8, and S₁₅ = 15/2 × (100 − 42) = 435. A negative common difference just means the terms fall.

Example 3: which term is it?

Which term of 7, 11, 15, … is 99? Set the rule equal to 99 and solve: 4n + 3 = 99, so n = 24. It is the 24th term. If n came out as a fraction, 99 would not be in the sequence at all.

Example 4: finding the rule from terms

Find the rule for 5, 11, 17, 23, 29. The gaps are all 6, so d = 6 and uₙ = 5 + (n − 1) × 6 = 6n − 1. In Find rule mode, the calculator shows the gaps, names the sequence arithmetic and gives the next terms 35, 41 and 47.

Example 5: seats in a theatre

The front row of a theatre has 20 seats and each row has 2 more than the one in front. How many seats are in 25 rows?

There are 1,100 seats, and the back row has 20 + 24 × 2 = 68.

Check you have the formula:

The sum of 1 to 100

The sum of the whole numbers from 1 to 100 is an arithmetic series with a = 1, d = 1 and n = 100. Pairing the first and last terms gives 101 each time, and there are 50 pairs, so the total is 50 × 101 = 5,050. That is the formula Sₙ = n/2 × (a + l), and the calculator gives the same.

Arithmetic or geometric?

An arithmetic sequence adds the same amount each time; a geometric sequence multiplies by the same amount. Check the gaps: if they are equal, it is arithmetic. If the ratios are equal, it is geometric. An arithmetic series never has a sum to infinity, because the terms do not shrink towards zero, which is why the calculator says so. For geometric sequences, choose Geometric in the calculator.

Common mistakes

• Using n instead of n − 1. The nth term is a + (n − 1)d. The first term has no d added.
• Getting the sign of d wrong. For a decreasing sequence, d is negative.
• Mixing up the nth term and the sum. uₙ is one term; Sₙ adds the first n terms.
• Finding d the wrong way round. It is a term minus the term before it, not the other way.
• Calling a sequence arithmetic without checking every gap. 2, 4, 8 has gaps 2 and 4, so it is not arithmetic.

Practise sequences in a short timed paper

Arithmetic sequences come up in AP Precalculus and Pre-Algebra & Algebra 1 practice, and in A-level Maths: Algebra & Functions in the UK. Each daily paper is a short timed paper of 8 questions worth 25 to 35 marks, and the papers mix topics.

Read next

Maths · AlgebraArithmetic series formula: Sₙ = n/2(2a + (n − 1)d), proofThe arithmetic series formula Sₙ = n/2[2a + (n − 1)d] and Sₙ = n/2(a + l): the proof, sigma notation, finding n, two-condition problems and examples.