How To Find The Sum Of An Arithmetic Series Calculator

An arithmetic series adds the terms of a sequence that changes by a fixed amount each step. The sum has a closed form, which means adding a thousand terms costs the same effort as adding five.

First term (a₁)Common difference (d)Keep the minus sign for a decreasing series. Zero makes every term identical.Number of terms (n)Count both ends: from term 4 to term 17 there are 14 terms.

Sum of all n terms

670

Last term (aₙ): 62
Average term (sum ÷ n): 33.5

Values update as you type. This calculator covers the single scenario its formula assumes — see Common Mistakes for what it leaves out.

The formula this calculator uses

S = n/2 x (a1 + an)

a1: First term of the series
d: Common difference added at each step
n: Number of terms being added
an: Last term, equal to a1 + (n - 1)d
S: Sum of the n terms

How to check the result by hand

1
Confirm the step is constant
Subtract consecutive terms in two or three places. If those differences differ, the series is not arithmetic and none of these formulas apply — check for a percentage pattern instead.
2
Record a1, d and n
a1 is the first term, d is the difference (keep its sign), and n is how many terms you are adding. Most errors trace back to one of these three being misread.
3
Find the last term
an = a1 + (n - 1)d. Here that is 5 + 19 x 3 = 62. Note the 19: with 20 terms, only 19 steps have happened.
4
Average the ends and scale by n
(a1 + an)/2 = (5 + 62)/2 = 33.5, which is also the average of every term. Multiply by 20 terms to get 670.
5
Cross-check against a small explicit sum
Add the first five terms by hand — 5 + 8 + 11 + 14 + 17 = 55 — and confirm the formula gives n/2 x (5 + 17) = 55 for the same five. Agreement means the setup is right.

For worked examples, common mistakes and the limits of this formula, read the full How To Find The Sum Of An Arithmetic Series page.