Arithmetic Sequence Calculator
Calculate any term of an arithmetic sequence, the sum of any range of terms, and instantly reverse-solve for a missing first term, common difference, or term count. Choose your solve mode, fill in the values you know, and get aₙ, Sₙ, and a full sequence list with show-your-work steps.
Formula
Worked example
With a₁ = 3, d = 5, n = 10: a₁₀ = 3 + (10 - 1)x5 = 3 + 45 = 48. Then S₁₀ = 10/2 x (3 + 48) = 5 x 51 = 255. To sum only terms 3 through 7: a₃ = 13, a₇ = 33, so S(3..7) = S(7) - S(2) = 7/2x(3+33) - 2/2x(3+8) = 126 - 11 = 115.
What an arithmetic sequence is
An arithmetic sequence is an ordered list of numbers in which each term differs from the previous one by a fixed amount called the common difference, written d. Starting from the first term a₁, you reach the next term by adding d, then add d again for the term after that, and so on. Because the gap between neighbours never changes, the sequence grows or shrinks in a perfectly straight line when plotted, which is why these sequences are sometimes described as linear. A positive d gives an increasing sequence, a negative d a decreasing one, and d = 0 produces a constant sequence where every term is equal to a₁.
Finding the nth term without listing them all
To reach the nth term you start at a₁ and add the common difference n - 1 times, because the first term needs no additions, the second needs one, and so on. That gives the closed formula aₙ = a₁ + (n - 1)d. The formula lets you jump straight to the 1,000th term without writing out the 999 terms before it. If d is negative the sequence counts downward, and a₁ is recovered by setting n = 1. To reverse-solve: if you know aₙ and n but not d, rearrange to d = (aₙ - a₁) / (n - 1). If you know aₙ, d, and n but not a₁, use a₁ = aₙ - (n - 1)d. To find which term holds a given value, compute n = (target - a₁) / d + 1 and verify it is a positive whole number.
Summing terms with the Gauss trick
The sum of the first n terms, Sₙ, has a neat shortcut. Pair the first term with the last, the second with the second-to-last, and so on: every pair adds to the same total, a₁ + aₙ. With n terms you get n/2 such pairs, so Sₙ = n/2 x (a₁ + aₙ). This is the trick the young Carl Friedrich Gauss reportedly used to add 1 through 100 in seconds, arriving at 50 pairs of 101 for a total of 5,050. For a partial sum from term j to term k, compute S(k) - S(j-1), the difference of two prefix sums, which this calculator handles automatically when you switch to the partial sum mode.
Reading the sequence list and chart
Turn on the sequence list to see every term and its running sum in a scrollable table, up to 100 terms. The chart plots both the individual term values and the cumulative sum as two series so you can immediately see the linear growth (or decline) of the terms and the parabolic growth of the sum. These visuals are especially useful for classroom demonstrations and for spotting errors when the sequence does not behave as expected.
Quick-reference: arithmetic sequence formulas
| Goal | Known values | Formula |
|---|---|---|
| Find nth term (aₙ) | a₁, d, n | aₙ = a₁ + (n - 1)d |
| Find sum (Sₙ) | a₁, d, n | Sₙ = n/2 x (a₁ + aₙ) |
| Find common difference (d) | a₁, aₙ, n | d = (aₙ - a₁) / (n - 1) |
| Find first term (a₁) | d, aₙ, n | a₁ = aₙ - (n - 1)d |
| Find position (n) | a₁, d, aₙ | n = (aₙ - a₁) / d + 1 |
| Partial sum (j to k) | a₁, d, j, k | S(j,k) = S(k) - S(j-1) |
Standard formulas covering all four reverse-solve modes.
Frequently asked questions
What is the common difference?
The common difference d is the constant amount added to one term to get the next. Find it by subtracting any term from the immediately following term, for example a₂ - a₁. A positive d makes the sequence rise, a negative d makes it fall, and zero keeps every term the same as a₁.
How do I find the nth term of an arithmetic sequence?
Use aₙ = a₁ + (n - 1)d. Plug in the first term, the common difference, and the position n you want. You add d exactly n - 1 times because the first term requires no additions. This calculator also lets you reverse-solve: if you know the nth term value but not d or a₁, select the matching solve mode.
How do I find the sum of an arithmetic sequence?
For the first n terms use Sₙ = n/2 x (a₁ + aₙ). For a partial sum from term j to term k, compute S(k) - S(j-1). Both are handled automatically by selecting "Sum of first n terms" or "Partial sum from term j to term k" in the solve mode dropdown.
How do I find which term has a specific value?
Select "Number of terms n" in the solve mode, enter a₁, d, and the target value. The calculator computes n = (target - a₁) / d + 1 and confirms the result is a positive whole number. If it is not, the target value does not appear in the sequence.
What is the difference between an arithmetic and a geometric sequence?
An arithmetic sequence adds a fixed amount between terms, so it changes in a straight line. A geometric sequence multiplies by a fixed ratio between terms, so it grows or decays exponentially. Use this calculator for the additive case and a geometric sequence calculator for the multiplicative one.