Arithmetic sequence calculator
Choose what you know – the first term and the common difference, or two terms such as a3 = 11 and a8 = 26 – then enter n. You get the nth term, the sum of the first n terms, the first term, the difference and a list of the first terms.
Runs in your browser: nothing you enter is sent anywhere.
How to use it
- Pick whether you know the first term and difference, or two terms.
- Fill in the matching boxes. The other boxes are ignored.
- Enter n, the position you need and the number of terms to add up.
How it works
The nth term is a₁ + (n − 1)d and the sum of the first n terms is n(a₁ + aₙ)/2. With two terms aₚ = u and a_q = v, the difference is d = (v − u)/(q − p) and the first term is u − (p − 1)d. Results show up to 12 significant digits.
Worked example
Take a3 = 11, a8 = 26 and n = 6.
- Difference: (26 − 11) / (8 − 3) = 3
- First term: 11 − 2 × 3 = 5
- Terms: 5, 8, 11, 14, 17, 20
- 6th term 20 and sum 6 × (5 + 20) / 2 = 75
Tips
- A negative difference gives a decreasing sequence.
- A difference of 0 repeats the same number.
- Adding the first and last terms is the same as adding the second and second-to-last: every such pair has the same total.
Frequently asked questions
What is an arithmetic sequence?
A sequence where each term is the previous one plus the same number, the common difference d. For example 5, 8, 11, 14 has a first term of 5 and d = 3.
What is the formula for the nth term?
The nth term is a₁ + (n − 1) × d. With a₁ = 5 and d = 3 the 6th term is 5 + 5 × 3 = 20.
How do I find the sum of an arithmetic sequence?
Add the first and last terms, multiply by the number of terms and divide by 2, so S = n × (a₁ + aₙ) / 2. For 5, 8, 11, 14, 17, 20 that is 6 × 25 / 2 = 75.
How do I find the sequence from two terms?
The difference is (value₂ − value₁) divided by (position₂ − position₁). Then step back from either term to position 1. If a3 = 11 and a8 = 26, d = 15 / 5 = 3 and a₁ = 11 − 2 × 3 = 5.