Arithmetic Sequence Calculator

The arithmetic sequence formulas

An arithmetic sequence has a constant gap, the common difference d, between consecutive terms. Two formulas do all the work:

nth term:  a_n = a₁ + (n − 1) · d
sum:       S_n = n/2 · (2·a₁ + (n − 1) · d)

Here a₁ is the first term, d is the amount added each step (it can be negative for a decreasing sequence), and n is how many terms you are counting. The sum formula is just the average of the first and last term, multiplied by how many terms there are.

A worked example

Take a₁ = 2 and d = 3. The sequence is 2, 5, 8, 11, 14, 17, 20, 23, 26, 29 …

To find the 10th term:

a₁₀ = 2 + (10 − 1) · 3 = 2 + 27 = 29

To add the first 10 terms:

S₁₀ = 10/2 · (2·2 + 9·3) = 5 · (4 + 27) = 5 · 31 = 155

So the 10th term is 29 and the running total is 155.

Terms, differences and partial sums

Notice each term rises by exactly d = 3, the signature of an arithmetic (not geometric) progression. The partial sum Sₙ grows faster than the terms themselves because every step adds the whole running line, not just the latest value.

A quick sanity check: the sum equals the number of terms times the average of the first and last term. For n = 10 that is 10 · (2 + 29) ÷ 2 = 10 · 15.5 = 155, which matches the table exactly.

Common pitfalls

• Off-by-one on n. The formula uses (n − 1)·d, not n·d. The first term has zero differences added, so a₁ = 2, not 5.
• Confusing arithmetic with geometric. Arithmetic sequences add a fixed d; geometric sequences multiply by a fixed ratio. If your gaps keep doubling, you need a geometric tool instead.
• Negative differences are fine. A common difference of d = −4 gives a falling sequence; the same formulas still apply.