Arithmetic Sequence Calculator
Enter the first term and the common difference to get any term of an arithmetic sequence and the sum of the first n terms. Or give two terms you already know and the calculator works backwards to a₁ and d. Fractions and decimals stay exact, term numbers go up to 10¹⁵, and every formula is shown with your numbers in it. Free, no sign-up.
Add this each time. Negative for a decreasing sequence.
A whole number from 1 up to 10¹⁵.
Results
79
Term a₂₀
820
Sum S₂₀ of the first 20 terms
4
Common difference
3
First term
d is positive, so the sequence increases.
General formula
aₙ = 4n − 1
Working
a₂₀ = a₁ + (20 − 1)d = 3 + (20 − 1) × 4 = 79
S₂₀ = 20(a₁ + a₂₀) ÷ 2 = 20 × (3 + 79) ÷ 2 = 820
The sequence
3, 7, 11, 15, 19, 23, 27, 31, …, 79
How to Use the Arithmetic Sequence Calculator
- 1. Choose what you know. First term and common difference is the usual case. Two terms of the sequence is for questions such as 'the 3rd term is 11 and the 8th is 36'. Find where a value sits tells you which term equals a number.
- 2. Fill in the fields. Whole numbers, decimals and fractions such as 3/4 all work, and the common difference may be negative. Positions are counted from 1.
- 3. Read the cards. You get the term aₙ, the sum Sₙ of the first n terms, the common difference and the first term.
- 4. Follow the working. Each formula is written out with your values substituted, followed by the first few terms of the sequence.
The Formulas
An arithmetic sequence adds the same amount, the common difference d, to get from each term to the next: a₁, a₁ + d, a₁ + 2d, …. The n-th term is therefore
aₙ = a₁ + (n − 1)d Sₙ = n(a₁ + aₙ) ÷ 2 = n(2a₁ + (n − 1)d) ÷ 2
The sum formula says that the total of an arithmetic series is the number of terms times the average of the first and last terms. The story goes that the young Carl Friedrich Gauss used exactly this to add 1 + 2 + … + 100: there are 100 terms with an average of (1 + 100) ÷ 2 = 50.5, so the sum is 5050. Pairing the terms from both ends — 1 + 100, 2 + 99, 3 + 98 — gives 50 pairs that each add to 101.
Worked Examples
- Term and sum. For 3, 7, 11, … we have a₁ = 3 and d = 4. The 20th term is a₂₀ = 3 + 19 × 4 = 79, and the sum of the first 20 terms is S₂₀ = 20 × (3 + 79) ÷ 2 = 820. As a formula, aₙ = 4n − 1.
- From two terms. If a₃ = 11 and a₈ = 36, the five steps from term 3 to term 8 add 25, so d = 25 ÷ 5 = 5. Then a₁ = 11 − 2 × 5 = 1. The sequence is 1, 6, 11, 16, …, with a₂₀ = 96 and S₂₀ = 970.
- Where does a value sit? In 3, 7, 11, … which term is 79? The number of steps from 3 to 79 is (79 − 3) ÷ 4 = 19, so it is term 19 + 1 = 20. The value 80 is not a term, because 77 ÷ 4 is not a whole number.
| You know | Use |
|---|---|
| a₁, d and n | aₙ = a₁ + (n − 1)d |
| a₁, aₙ and d | n = (aₙ − a₁) ÷ d + 1 |
| Two terms aₚ and a_q | d = (a_q − aₚ) ÷ (q − p), then a₁ = aₚ − (p − 1)d |
| a₁, aₙ and n | Sₙ = n(a₁ + aₙ) ÷ 2 |
| Any three consecutive terms a, b, c | b = (a + c) ÷ 2 — each term is the mean of its neighbours |
Frequently Asked Questions
- What is the formula for the nth term of an arithmetic sequence?
- aₙ = a₁ + (n − 1)d, where a₁ is the first term, d is the common difference and n is the position. For 5, 8, 11, … it gives aₙ = 5 + 3(n − 1) = 3n + 2, so the 50th term is 152.
- How do you find the sum of an arithmetic series?
- Multiply the number of terms by the average of the first and last: Sₙ = n(a₁ + aₙ) ÷ 2. If you do not know the last term, use Sₙ = n(2a₁ + (n − 1)d) ÷ 2. Both give 820 for the first 20 terms of 3, 7, 11, ….
- How do I find the common difference?
- Subtract any term from the one after it. If you only know two terms that are not neighbours, divide their difference by the number of steps between them: with a₃ = 11 and a₈ = 36, d = (36 − 11) ÷ (8 − 3) = 5.
- How do I find the number of terms?
- Use n = (last term − first term) ÷ d + 1. For 7, 12, 17, …, 102, that is (102 − 7) ÷ 5 + 1 = 20. If the result is not a whole number, the last term you were given is not actually in the sequence.
- Can the common difference be zero, negative or a fraction?
- Yes to all three. With d = 0 every term is the same; with a negative d the sequence decreases, as in 20, 17, 14, …; with a fraction such as d = 1/2 you get 2, 2.5, 3, …. The calculator handles them exactly.
- What is the difference between an arithmetic sequence and an arithmetic series?
- The sequence is the list of terms (3, 7, 11, 15); the series is the sum of them (3 + 7 + 11 + 15 = 36). This tool gives you both: the n-th term and the partial sum Sₙ.
- Is an arithmetic progression the same thing?
- Yes. Arithmetic progression, often written AP, is another name for an arithmetic sequence, and 'arithmetic mean' is the same idea in reverse: the middle term of three consecutive terms is the average of the other two.