Enter your sequence's first term (a₁), common difference (d), and term index (n) to find the nth term and the sum of the first n terms of any arithmetic sequence. You can also solve for the common difference from two known terms. Results include a breakdown chart and a terms table showing each value in the sequence.
nth Term (aₙ)
--
Sum of First n Terms (Sₙ)
--
Common Difference (d)
--
First Term (a₁)
--
Term Index (n)
--
An arithmetic sequence is an ordered list of numbers where each term after the first is obtained by adding a constant value — called the common difference (d) — to the previous term. For example, 2, 5, 8, 11, 14 is an arithmetic sequence with a common difference of 3.
Use the formula aₙ = a₁ + (n − 1) × d, where a₁ is the first term, d is the common difference, and n is the term's position. For example, the 10th term of the sequence starting at 2 with d = 3 is a₁₀ = 2 + (10 − 1) × 3 = 29.
The sum of the first n terms (arithmetic series) is given by Sₙ = n/2 × (2a₁ + (n − 1)d), or equivalently Sₙ = n/2 × (a₁ + aₙ). Simply enter your values into the calculator above and the sum is computed automatically.
If you know two terms and their positions, use the formula d = (aₖ₂ − aₖ₁) / (k₂ − k₁). For instance, if the 2nd term is 5 and the 6th term is 17, then d = (17 − 5) / (6 − 2) = 3. Select 'Find common difference' mode in the calculator to solve this automatically.
An arithmetic sequence is the ordered list of terms (e.g., 2, 5, 8, 11…), while an arithmetic series is the sum of those terms (e.g., 2 + 5 + 8 + 11 = 26). This calculator computes both simultaneously.
In an arithmetic sequence, a constant value (d) is added between terms. In a geometric sequence, a constant ratio (r) is multiplied between terms. For example, 2, 4, 6, 8 is arithmetic (d=2), while 2, 4, 8, 16 is geometric (r=2).
Subtract any term from the next term throughout the sequence. If the difference is always the same constant value, the sequence is arithmetic. For example, in −12, −1, 10, 21, the differences are all 11, confirming it is arithmetic with d = 11.
Yes. A negative common difference means the sequence is decreasing. For example, 20, 15, 10, 5, 0, −5 has a common difference of −5. The same formulas apply regardless of whether d is positive, negative, or even zero.