What is the formula for the nth term of an arithmetic sequence?

aₙ = a₁ + (n − 1)d, where a₁ is the first term and d is the common difference. For 3, 7, 11, 15, …, a₁ = 3 and d = 4, so aₙ = 3 + 4(n − 1) = 4n − 1 and the 10th term is 39.

How do you find the sum of an arithmetic sequence?

Use Sₙ = n(a₁ + aₙ)/2: the number of terms times the average of the first and last term. The first 10 terms of 3, 7, 11, … end at 39, so S₁₀ = 10(3 + 39)/2 = 210.

How do you find the common difference?

Subtract any term from the one after it. In 50, 46, 42, 38 the difference is 46 − 50 = −4. If you only know two terms that aren't next to each other, divide their difference by the number of steps between them.

How do you find the first term from two other terms?

Find d first, then step back. If a₄ = 17 and a₉ = 37, five steps add 20, so d = 4, and a₁ = 17 − 3 × 4 = 5. Choose 'From two terms' in the calculator to see this worked out.

How do you know which term a number is?

Set aₙ equal to the number and solve for n. In 5, 8, 11, …, 62 = 5 + 3(n − 1) gives n = 20, so 62 is the 20th term. If n comes out as a fraction or a negative number, the value isn't in the sequence.

What is the recursive formula for an arithmetic sequence?

State the first term and the rule for getting each term from the previous one: a₁ = 3, aₙ = aₙ₋₁ + 4. It describes the same sequence as the explicit formula 4n − 1, but you have to work through every earlier term to reach a far-off one.

Arithmetic Sequence Calculator: nth Term and Sum

An arithmetic sequence adds the same amount every time: 3, 7, 11, 15, and so on, up by 4 each step. Finding the 10th term is easy enough to do in your head. The 250th term, or the total of the first 80 terms, is where a formula earns its keep, and that’s what this calculator handles. It finds any term, adds up as many terms as you like, and writes the sequence’s explicit and recursive formulas, working from whatever you happen to know about the sequence.

Arithmetic sequence formulas: a sub n equals a sub 1 plus (n minus 1) d, and S sub n equals n times (a sub 1 plus a sub n) over 2 — The nth term and the sum of the first n terms

How to use the arithmetic sequence calculator

Pick what you know. “First term and difference” takes and . “From a list of terms” takes the start of the sequence, like 5, 8, 11, 14. “From two terms” takes any two terms and their positions.
Enter n, the term you want. The calculator also adds up the first terms.
Optionally, enter a value in “Find the position of” to see whether it’s in the sequence and where.
Read the results: , , both formulas, a plot of the first 12 terms and every step of the working.

The arithmetic sequence formulas

The first formula says that to reach term you start at and take steps of size . It’s , not , because the first term needs no steps at all. The second formula, the arithmetic series sum, multiplies the number of terms by the average of the first and last ones. Both work for any arithmetic sequence: increasing, decreasing, fractions or negatives.

Worked example: a₁ = 3, d = 4

These are the calculator’s default inputs.

Step 1 — the nth term formula.

Step 2 — the 10th term.

Step 3 — the sum of the first 10 terms.

You can check the sum by hand: 3 + 7 + 11 + 15 + 19 + 23 + 27 + 31 + 35 + 39 = 210. The simplified formula is worth noticing too. Its coefficient of is always , so the sequence behaves like the line , and its points sit on that line when you plot them.

Working from a list of terms

Type the first few terms and the calculator finds by subtracting neighbours, after checking that the differences really are all equal. With 5, 8, 11, 14, the difference is 3 and the formula is .

That makes “which term is 62?” a one-line equation:

So 62 is the 20th term. Try 60 instead and you get , not a whole number, so 60 isn’t in the sequence at all. If the differences aren’t constant, the calculator says so, and if the ratios are constant instead, it points you to the geometric version.

Working from two terms

Sometimes you’re given two terms that aren’t next to each other, like and , and asked for the formula.

The sequence 5, 9, 13, 17, 21, 25, 29, 33, 37 with a₄ = 17 and a₉ = 37 highlighted; five +4 steps join them, so d = 20 ÷ 5 = 4, and three −4 steps lead back from a₄ to a₁ = 17 − 3 × 4 = 5 — Count the steps between the two terms to get d, then walk back to a₁

From position 4 to position 9 is five steps, covering , so . Walking back three steps from gives , and the formula is . The calculator’s “From two terms” mode is set up with this example.

Decreasing sequences and fractions

Nothing changes when the difference is negative or a fraction, but these are the cases where mistakes creep in, so it’s worth seeing them once.

A decreasing sequence. For 50, 46, 42, 38, …, the difference is . The 15th term is , and the sum of the first 15 terms is

The terms pass zero between the 13th (2) and the 14th (−2), and the negative terms at the end eat into the total. That’s why the sum is smaller than you might guess from the first few terms.

Fractions. For , the difference is . The 7th term is , and the first seven terms add to . The calculator keeps these as fractions rather than rounding to 0.8333…, so the answers stay exact all the way through. Type the list as 1/2, 5/6, 7/6 to see it.

Arithmetic sequence formulas at a glance

To find	Formula	Example (, )
any term
common difference
sum of terms
sum without the last term
position of a value		39 is term 10
recursive form

Where arithmetic sequences show up

Anything that changes by a fixed amount per step is an arithmetic sequence. A theater with 22 seats in the first row and 3 more in each row after it has seats in row 12. A savings plan that puts away in month one and more each month reaches in month 12. A taxi fare with a flat charge plus a fixed price per mile is one too. In each case the sum formula answers the follow-up question: how many seats in total, or how much saved over the year.

Common mistakes

Using instead of . is , not .
Getting the sign of wrong. For 50, 46, 42, … the difference is . Always subtract a term from the one after it.
Dividing by the wrong gap. From to is 5 steps, not 9 or 4.
Assuming a pattern is arithmetic. 1, 4, 9, 16 has differences 3, 5, 7, so no single works.
Mixing up and . One is a single term; the other is a total.

For the full explanation, see the arithmetic sequences lesson, and when you want to test yourself, the arithmetic sequences practice problems check each answer as you go.

Arithmetic Sequence Calculator