Arithmetic Sequence Calculator
Arithmetic sequence nth term aₙ = a₁ + (n − 1)d and sum Sₙ of the first n terms, with the first 20 terms listed and the position of any value. How it works
Use the Arithmetic Sequence Calculator
About the Arithmetic Sequence Calculator
In the sequence 3, 7, 11, 15, … the 25th term is 99 and the first 25 terms add up to 1275. This arithmetic sequence calculator uses aₙ = a₁ + (n − 1)d and Sₙ = n(a₁ + aₙ)/2, lists the first 20 terms or any run of up to 50 (terms 10 to 13 of 1, 3, 5, … are 19, 21, 23 and 25) and finds which term equals a value you type. Given two terms instead, such as a₂ = 3 and a₆ = 11, it works out a₁ = 1 and d = 2.
Worked arithmetic sequences
| a₁ | d | n | aₙ | Sₙ | First terms |
|---|---|---|---|---|---|
| 1 | 2 | 10 | 19 | 100 | 1, 3, 5, 7, 9, … |
| 1 | 1 | 100 | 100 | 5050 | 1, 2, 3, 4, 5, … |
| 3 | 4 | 25 | 99 | 1275 | 3, 7, 11, 15, 19, … |
| 5 | 3 | 20 | 62 | 670 | 5, 8, 11, 14, 17, … |
| 20 | 2 | 15 | 48 | 510 | 20, 22, 24, 26, 28, … |
| 100 | -7 | 15 | 2 | 765 | 100, 93, 86, 79, 72, … |
| -3 | 0.5 | 12 | 2.5 | -3 | -3, -2.5, -2, -1.5, -1, … |
| 0.1 | 0.2 | 10 | 1.9 | 10 | 0.1, 0.3, 0.5, 0.7, 0.9, … |
Which term is it?
| a₁ | d | Value | Tool says |
|---|---|---|---|
| 1 | 2 | 99 | n = 50 |
| 5 | 3 | 62 | n = 20 |
| 100 | -7 | 2 | n = 15 |
| 7 | 5 | 152 | n = 30 |
| 1 | 2 | 50 | The target is not a term of this sequence (n would not be a whole number). |
| 5 | 3 | 2 | The target comes before the first term (n would be below 1). |
Frequently asked questions
- What is the formula for the nth term of an arithmetic sequence?
- aₙ = a₁ + (n − 1)d. For 5, 8, 11, …, a₁ = 5 and d = 3, so a₂₀ = 5 + 19 × 3 = 62.
- How do you find the sum of an arithmetic sequence?
- Sₙ = n(a₁ + aₙ)/2: the number of terms times the average of the first and last term. The sum of 1 to 100 is 100 × (1 + 100)/2 = 5050, and 5 + 8 + … + 62 is 20 × 67/2 = 670.
- How do you find which term a number is?
- Solve n = (value − a₁)/d + 1. In 1, 3, 5, …, 99 is term (99 − 1)/2 + 1 = 50. If the result is not a whole number, the value is not in the sequence: 50 would be term 25.5, so it is not a term.
- How do you find the common difference from two terms?
- Divide the difference of the terms by the difference of their positions. If a₃ = 11 and a₇ = 23, then d = (23 − 11)/(7 − 3) = 3 and a₁ = 11 − 2 × 3 = 5. Choose Two known terms and type positions 3 and 7 with the values 11 and 23: the tool shows a₁ = 5 and d = 3, and with n = 7 it shows 23.
- Can the common difference be negative or a decimal?
- Yes. 100, 93, 86, … has d = -7, a₁₅ = 2 and S₁₅ = 765. With a₁ = 0.1 and d = 0.2 the terms are 0.1, 0.3, 0.5, …, a₁₀ = 1.9 and S₁₀ = 10 exactly.
Related Tools
View all toolsBasic Calculator
Basic calculator for + − × ÷ in the standard order of operations (2 + 3 × 4 = 14), with % and memory keys, ^ for powers and the result as a fraction.
Fibonacci Sequence Generator
Fibonacci numbers from any two starting values: the first 1 to 1000 terms, the exact nth term up to n = 10,000, and the golden ratio estimate.
LCM and GCD Calculator
GCD (also called GCF or HCF) and LCM of 2 to 10 whole numbers, with the Euclidean steps and the prime factors of each number shown.
Decimal to Fraction
Convert a decimal to a fraction in lowest terms and see the working: 0.75 = 3/4, 2.75 = 11/4 or 2 3/4, and 0.333… (3 repeating) = 1/3.
Derivative Calculator
Enter f(x) and a point x = a to get f′(a), f(a), the tangent line and its graph, plus the formula for f′(x). Flags corners, jumps and vertical tangents.
Determinant Calculator
Determinant of a square matrix up to 10 × 10 as an exact fraction and a decimal, with the cofactor working shown for 2 × 2 and 3 × 3.