Arithmetic Sequence Calculator

What is an arithmetic sequence?

An arithmetic sequence is a list of numbers in which each term is the one before it plus a fixed amount. That amount is the common difference, written d. In 2, 5, 8, 11, 14 the common difference is 3. In 20, 17, 14, 11 it is −3, because each step takes 3 away. The terms are numbered from 1: a₁ is the first term, a₂ the second, and aₙ means the term in position n.

British and Indian textbooks call the same thing an arithmetic progression, or AP. Adding the terms together gives an arithmetic series, so 2 + 5 + 8 + 11 is the series that belongs to the sequence 2, 5, 8, 11.

The nth term formula

To reach term n you start at a₁ and add d once for every step after the first. There are n − 1 of those steps, which is why the formula says n − 1 and not n. Here is the method for the 52nd term of −24, −7, 10, …:

1. Find d: −7 − (−24) = 17. Check the next pair too: 10 − (−7) = 17.
2. Note the first term and the position: a₁ = −24 and n = 52.
3. Substitute: a₅₂ = −24 + (52 − 1) × 17 = −24 + 867 = 843.

Show the 52nd term of −24, −7, 10

The explicit formula

Multiply out the brackets and you get a formula with n on its own: aₙ = d × n + (a₁ − d). For 1, 6, 11, … that is aₙ = 5n − 4. Put in n = 1 and you get 1; put in n = 3 and you get 11. When a question asks for the explicit formula, the general term or a rule for the nth term, this is the form it wants.

Show the formula for 1, 6, 11

The number in front of n is always the common difference. That makes an arithmetic sequence a straight-line function of n with d as its slope, and you can plot the terms as points on a line. The rate of change calculator works out that slope for any two points.

The recursive formula

A recursive formula gives the first term and says how to get each term from the one before: a₁ = 9, aₙ = aₙ₋₁ − 5. It is short to write but slow for far-off terms, because you have to walk through each one. To find a₆ you can count 9, 4, −1, −6, −11, −16. Or turn it into the explicit formula aₙ = −5n + 14 and get a₆ = −30 + 14 = −16 in one line.

Show a₆ for a₁ = 9, aₙ = aₙ₋₁ − 5

The table shows the same rule worked out for a few sequences. Each formula is checked against the first five terms.

How to find the common difference

Subtract any term from the term right after it. Always take the later term minus the earlier one, or the sign comes out wrong. For 170, 85, 0, −85 the difference is 85 − 170 = −85, so the sequence drops by 85 each time and its formula is aₙ = −85n + 255.

Show the formula for 170, 85, 0, −85

Check more than one pair. If the gaps are not all equal, the list is not arithmetic and the formulas on this page do not apply. 3, 6, 12, 24 has gaps of 3, 6 and 12; each term doubles, so it is a geometric sequence. 1, 4, 9, 16 has gaps of 3, 5 and 7, which grow by 2 each time. That is a quadratic sequence. The calculator tells you which of these it finds.

Test 3, 6, 12, 24 in the calculator

The difference can be a decimal or a fraction. 0.5, 0.75, 1, 1.25 has d = 0.25. When the step is a third, the calculator keeps it as 1/3 instead of 0.333, so the 100th term stays exact and you do not pick up rounding errors along the way.

Finding the sequence from two terms

Many problems give two terms from the middle, such as "a₃ = 5 and a₇ = 13, find a₁₁". From a₃ to a₇ there are 7 − 3 = 4 steps, and the value rises by 13 − 5 = 8, so each step is d = 8 ÷ 4 = 2. Step back from a₃ to the first term: a₁ = 5 − 2 × 2 = 1. Now a₁₁ = 1 + 10 × 2 = 21.

Show a₁₁ from a₃ = 5 and a₇ = 13

The same two lines answer "find the first term when a₃ = 25 and a₁₉ = 105": d = 80 ÷ 16 = 5 and a₁ = 25 − 2 × 5 = 15. When the values do not divide evenly, d is a fraction, and the calculator keeps it as one.

Find a₁ from a₃ = 25 and a₁₉ = 105

Sum of an arithmetic sequence

Use the first form when you know the last term. The second one does not need it, because 2a₁ + (n − 1) × d is just a₁ + aₙ written out. Both give the same answer.

The formula comes from pairing terms. Write the terms in order, then write them again backward underneath. Each column adds up to the same total, the first term plus the last. There are n columns, so the two rows hold n × (a₁ + aₙ), and one row is half of that. A well-known story says the young Carl Friedrich Gauss added 1 to 100 this way in his head: 50 pairs that each make 101, so 5,050. Historians doubt the details of the story, and the method works for any arithmetic sequence. The calculator draws these pairs under each result.

For the sum of the first 35 terms of 3, 11, 19, …, the common difference is 8. The 35th term is 3 + 34 × 8 = 275, and the sum is 35 × (3 + 275) ÷ 2 = 4,865.

Add the first 35 terms of 3, 11, 19

A few sums come up again and again in homework and puzzles:

How many terms are in a sequence?

If you know the first term, the last term and the step, turn the nth term formula around: n = (aₙ − a₁) ÷ d + 1. The + 1 counts the first term, which people often forget. From 3 to 297 in steps of 3 there are (297 − 3) ÷ 3 + 1 = 99 terms. If the result is not a whole number, the last number is not in the sequence. The "Which term is it?" box under the results does this for any number you type.

Count the terms from 3 to 297

Arithmetic vs geometric sequences

Both kinds start from a first term and repeat one fixed step. An arithmetic sequence adds the same number each time, and a geometric sequence multiplies by the same number.

Money shows the difference well. Simple interest adds the same amount every year, so the balance is an arithmetic sequence. Compound interest adds a percentage of a growing balance, so the balance is geometric and pulls ahead over time. The interest calculator works out both for real amounts, and the compound interest calculator follows compounding year by year.

Common mistakes

• Writing n where the formula has n − 1. The 10th term of 2, 5, 8 is 2 + 9 × 3 = 29, not 2 + 10 × 3 = 32.
• Subtracting in the wrong order. d is the later term minus the earlier one, so 20, 17, 14 has d = −3.
• Mixing up position and value. a₅ = 20 says the fifth term is 20: the 5 is where it sits and the 20 is its value.
• Leaving out the + 1 when counting terms. 3, 6, …, 297 has 99 terms, not 98.
• Using the formulas on a list that is not arithmetic. Check two or three gaps before you start.

Where arithmetic sequences show up

Anything that changes by the same amount at each step follows this pattern. A theatre with 20 rows that starts with 15 seats and adds 2 seats per row has 53 seats in the back row and 680 seats in total. Saving $50 a week gives a balance of $50, $100, $150 and so on. House numbers on one side of a street run 1, 3, 5, …, and a taxi fare with a fixed start price plus the same amount per mile goes up by the same step for every mile.

Count the seats in the 20-row theatre