Arithmetic Sequence Calculator
Find the nth term, common difference, sum and missing terms of an arithmetic sequence, with exact fractions and step-by-step working.
Examples
The 10th term
- First term a₁
- 3
- Common difference d
- 4 (increasing)
- General term
- aₙ = 3 + (n − 1)(4) = 4n − 1
- Sequence
- 3, 7, 11, 15, 19, …
Step-by-Step Solution
Step 1 — Identify the values
a₁ = 3
d = 4
n = 10
Step 2 — Use the nth-term formula
aₙ = a₁ + (n − 1)d
Step 3 — Substitute n = 10
a₁₀ = 3 + (10 − 1)(4)
a₁₀ = 3 + 9 × 4
Step 4 — Calculate
a₁₀ = 3 + 36
a₁₀ = 39
Final answer: a₁₀ = 39
Terms
| n | aₙ | Decimal |
|---|---|---|
| a₁ | 3 | |
| a₂ | 7 | |
| a₃ | 11 | |
| a₄ | 15 | |
| a₅ | 19 | |
| a₆ | 23 | |
| a₇ | 27 | |
| a₈ | 31 | |
| a₉ | 35 | |
| a₁₀ | 39 |
Chart of n against aₙ
The points lie on a straight line with slope d = 4: each step to the right adds the same amount.
Formula reference
| Find | Formula | Use it when |
|---|---|---|
| nth term | aₙ = a₁ + (n − 1)d | Any term from the first term and the common difference |
| Common difference | d = aₙ − aₙ₋₁ | Any term minus the one before it; must be the same every time |
| Sum of n terms | Sₙ = n/2 [2a₁ + (n − 1)d] | When you know a₁, d and n |
| Sum of n terms | Sₙ = n(a₁ + aₙ)/2 | When you know the first and last terms |
| Terms from m to n | n − m + 1 terms | Both ends count |
| Arithmetic mean | A = (a + b)/2 | The middle term of a, A, b |
| k means between a and b | d = (b − a)/(k + 1) | The sequence has k + 2 terms |
If each term is multiplied by the same number instead, use the Geometric Sequence Calculator; to add up arithmetic or geometric terms, try the Sequence Sum Calculator. The arithmetic mean of a whole data set is on the Average Calculator, and constant percentage growth (a geometric pattern) on the Percentage Calculator. An arithmetic sequence is a linear function of n, which the Slope Calculator explores, and the Scientific Calculator handles quick arithmetic.
This arithmetic sequence calculator (arithmetic progression calculator) works with sequences where the same number is added each time, like 3, 7, 11, 15, … It finds the common difference, any nth term, the sum of the first n terms or of any block of terms, fills in missing terms, inserts arithmetic means between two numbers, and generates the terms.
Every result comes with step-by-step working, a term table and a chart. Fractions and decimals stay exact, and if the terms you enter do not have a constant difference, the calculator tells you the sequence is not arithmetic.
Worked Calculation Examples
| Scenario | Result | Calculation Step |
|---|---|---|
| 3, 7, 11, 15, … find a₁₀ | a₁₀ = 39 | a₁ = 3 and d = 4, so a₁₀ = 3 + (10 − 1)(4) = 3 + 36 = 39. |
| 5, 10, 15, 20, … find S₂₀ | S₂₀ = 1,050 | a₂₀ = 5 + 19(5) = 100, so S₂₀ = 20(5 + 100)/2 = 1,050. |
| 20, 17, 14, 11, … find a₁₅ | a₁₅ = −22 | d = 17 − 20 = −3, so a₁₅ = 20 + 14(−3) = 20 − 42 = −22. |
| 4, __, 12, 16: find the missing term | 8 | The terms 12 and 16 give d = 4 (and 12 − 4 = 2d confirms it), so the missing term is 4 + 4 = 8. |
| Insert 3 arithmetic means between 5 and 25 | 5, 10, 15, 20, 25 | There are 3 + 2 = 5 terms, so d = (25 − 5)/4 = 5. |
| Is 2, 6, 10, 15 arithmetic? | No | The differences are 4, 4 and 5. They are not all equal, so there is no common difference and the sequence is not arithmetic. |
What Is an Arithmetic Sequence?
An arithmetic sequence (or arithmetic progression) is a list of numbers in which each term is the previous term plus a fixed number, the common difference d. In 2, 5, 8, 11, 14, … each term is 3 more than the one before, so d = 3.
A sequence is the list of terms, a₁, a₂, a₃, … A series is the sum of terms, a₁ + a₂ + a₃ + … The calculator finds terms of the sequence and sums of the series; they are different questions.
Arithmetic Sequence Formula
The nth term is aₙ = a₁ + (n − 1)d. To reach the nth term you start at a₁ and add d once for each step, and there are n − 1 steps from term 1 to term n.
For 3, 7, 11, 15, …: a₁ = 3 and d = 4, so aₙ = 3 + (n − 1)4 = 4n − 1. The 10th term is 4(10) − 1 = 39.
How to Find the Common Difference
Subtract any term from the term after it: d = aₙ − aₙ₋₁. For 5, 9, 13, 17: 9 − 5 = 4, 13 − 9 = 4 and 17 − 13 = 4, so d = 4.
Check every pair, not just the first. For 4, 9, 15, 20 the differences are 5, 6 and 5, so it is not an arithmetic sequence at all, and there is no common difference to find.
How to Find the nth Term
Identify a₁ and d, then substitute into aₙ = a₁ + (n − 1)d. For 4, 7, 10, 13, … the 15th term is a₁₅ = 4 + (15 − 1)(3) = 4 + 42 = 46.
If you know two terms but not a₁, subtract them to find d first. If a₃ = 10 and a₈ = 25, then 5d = 25 − 10, so d = 3 and a₁ = 10 − 2(3) = 4.
How to Find the Sum of an Arithmetic Sequence
The sum of the first n terms is Sₙ = n(a₁ + aₙ)/2: the number of terms times the average of the first and last terms. Equivalently, Sₙ = n/2 [2a₁ + (n − 1)d].
For 2, 5, 8, … with n = 10: a₁₀ = 2 + 9(3) = 29, so S₁₀ = 10(2 + 29)/2 = 155. To add a block from term m to term n, find aₘ and aₙ, count n − m + 1 terms, and use the same formula.
How to Find a Missing Term
Use two known terms to find d. In 2, 5, __, 11 the first two terms give d = 3, so the missing term is 5 + 3 = 8, and 8 + 3 = 11 confirms it.
When the known terms are not next to each other, divide by the gap in positions. In 10, __, 20, 25, the terms 10 and 20 are two positions apart, so 2d = 10 and d = 5; the missing term is 15. A single known term is not enough: any d would work.
Arithmetic Mean
The arithmetic mean of two numbers is A = (a + b)/2. It is the middle term of the arithmetic sequence a, A, b: the mean of 8 and 20 is 14, and 8, 14, 20 has d = 6.
To insert k arithmetic means between a and b, build a sequence with k + 2 terms from a to b. Its common difference is d = (b − a)/(k + 1). Inserting 3 means between 4 and 20 gives d = 16/4 = 4 and the sequence 4, 8, 12, 16, 20.
Increasing vs Decreasing Arithmetic Sequences
If d > 0 the terms increase (3, 7, 11, …); if d < 0 they decrease (20, 16, 12, 8, … has d = −4, so a₁₀ = 20 + 9(−4) = −16); and if d = 0 every term is the same (5, 5, 5, …), which still counts as arithmetic, with aₙ = 5 and Sₙ = 5n.
Arithmetic Sequence Examples
Seats in a theatre: 20 in the first row and 3 more in each row. Row 12 has 20 + 11(3) = 53 seats, and the first 12 rows hold 12(20 + 53)/2 = 438 seats.
Savings: $50 saved in week 1 and $10 more each week. In week 26 you save 50 + 25(10) = $300, and over 26 weeks 26(50 + 300)/2 = $4,550.
The odd numbers 1, 3, 5, … have d = 2; the sum of the first n odd numbers is n(1 + 2n − 1)/2 = n².
Arithmetic Sequence vs Geometric Sequence
An arithmetic sequence adds the same number each time: 2, 5, 8, 11 has common difference d = 3. A geometric sequence multiplies by the same number each time: 2, 6, 18, 54 has common ratio r = 3.
To tell them apart, compute the differences and the ratios of consecutive terms. Constant differences mean arithmetic; constant ratios mean geometric. Arithmetic sequences grow linearly, and their graph of n against aₙ is a straight line. Geometric sequences grow or shrink exponentially.
Common Mistakes
Using n instead of n − 1 in aₙ = a₁ + (n − 1)d. Subtracting in the wrong order, which flips the sign of d. Checking only the first difference and assuming the sequence is arithmetic. Averaging unequal differences and calling the result a common difference. Counting the terms from m to n as n − m instead of n − m + 1. Confusing a sequence (the list) with a series (the sum). Rounding fractions early and carrying the rounding error through the calculation.
How to Use the Arithmetic Sequence Calculator
- Choose what you want to do: find the nth term, work from given terms, find a sum, generate terms, fill in a missing term, or find arithmetic means.
- Enter the values. Numbers can be whole, negative, decimal or fractions such as 1/2. Separate terms with commas, and mark a missing term with __ or ?.
- Press Calculate, or tap one of the examples. Results stay exact (3/2 rather than 1.5), with a decimal alongside when helpful.
- Check the step-by-step solution. When you enter terms, every consecutive difference is shown, and the calculator tells you plainly if the sequence is not arithmetic.
- Use the term table and the chart of n against aₙ: the points of an arithmetic sequence always lie on a straight line with slope d.
Frequently Asked Questions
What is an arithmetic sequence?
A sequence where the difference between consecutive terms is always the same number, the common difference d. For example 2, 5, 8, 11, … has d = 3.
How do you find the common difference?
Subtract any term from the next one: d = aₙ − aₙ₋₁. Check every pair; if the differences are not all equal, the sequence is not arithmetic.
What is the nth-term formula?
aₙ = a₁ + (n − 1)d, where a₁ is the first term, d is the common difference and n is the term position.
How do you find the nth term of an arithmetic sequence?
Find a₁ and d, then substitute n. For 4, 7, 10, 13, …, a₁₅ = 4 + (15 − 1)(3) = 46.
How do you find the sum of an arithmetic sequence?
Use Sₙ = n(a₁ + aₙ)/2, or Sₙ = n/2 [2a₁ + (n − 1)d]. For a₁ = 2, d = 3 and n = 10, a₁₀ = 29 and S₁₀ = 10(2 + 29)/2 = 155.
How do you find a missing term?
Use two known terms to find the common difference, dividing their difference by how many positions apart they are, then add or subtract d to fill the gap. In 2, 5, __, 11, d = 3, so the missing term is 8.
What is the difference between an arithmetic and a geometric sequence?
An arithmetic sequence adds a constant difference each time (2, 5, 8, 11); a geometric sequence multiplies by a constant ratio (2, 6, 18, 54).
Can an arithmetic sequence have a negative common difference?
Yes. Then the terms decrease: 20, 16, 12, 8, … has d = −4, and its 10th term is 20 + 9(−4) = −16.
Can the common difference be zero?
Yes. A constant sequence like 5, 5, 5, 5 is arithmetic with d = 0. Every term equals a₁ and the sum of n terms is n·a₁.
What is the arithmetic mean?
The average of two numbers, A = (a + b)/2. It is the number that sits exactly halfway between them, so a, A, b is an arithmetic sequence. The mean of 8 and 20 is 14.
How do you insert arithmetic means between two numbers?
To insert k means between a and b, use d = (b − a)/(k + 1) and add d repeatedly from a. Inserting 3 means between 4 and 20 gives d = 4 and the sequence 4, 8, 12, 16, 20.
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Last updated: September 27, 2026.