Arithmetic series sum is one of 4 sequences formulas in the algebra section of this library, and it is used at high school level.
Why arithmetic series sum works
Pair the first term with the last, the second with the second last, and so on. Each pair adds to the same total, because whatever one term gains the other loses. There are n over 2 such pairs, so the whole sum is the count times the average of the first and last terms.
What each symbol means
$S_n$ is the sum and $a_1,a_n$ are the first and last terms.
Arithmetic series sum: when it holds
$n$ is a positive integer and terms must have constant difference.
When it stops applying
The pairing only balances when every gap is equal. Adding 1, 2, 4 and 7, whose gaps grow, the formula reports (4/2)(1 + 7) = 16, while the true total is 14. It still returns a number, so nothing warns you.
Arithmetic series sum: a worked example
$2+5+8+11+14=\frac{5}{2}(2+14)=40$.
The mistake to avoid
What people do: Using the position of the last term as n when the sum does not start at term 1.
Why it goes wrong: Adding terms 5 through 12 means adding eight numbers, not twelve. Using 12 inflates the answer by counting four terms that were never in the sum.
Do this instead: Compute the count as last position minus first position plus 1, so 12 - 5 + 1 gives the correct n of 8.
Arithmetic series sum: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $S_n$ is the sum and $a_1,a_n$ are the first and last terms.
- Check the conditions before substituting. $n$ is a positive integer and terms must have constant difference.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most algebra slips.
Where this formula fits
- Subject
- Algebra formulas — 28 entries in this library
- Topic
- Sequences
- Level
- High school
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where arithmetic series sum comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Sequences and Series — the lesson behind this formula: recognize arithmetic, geometric, and recursive patterns.
- Algebra Calculator — check your substitution and the value it produces.
- Study algebra — the subject guide that explains the ideas these formulas compress.
- Algebra I Practice — questions that make you retrieve the formula instead of recognising it.
- All 28 algebra formulas — the full grouped reference, or the complete formula library.
Questions about arithmetic series sum
What is the sum of the whole numbers from 1 to 100?
It is 50 pairs each worth 101, which is 5050. This is the classic schoolroom example, and the formula reproduces it as (100/2) times (1 + 100).
What if I do not know the last term?
Use the second version, which builds the last term from a1 and d. It needs only the first term, the common difference and how many terms you are adding.
Does it still work when the terms are negative?
Yes, and no special handling is needed. The pairing argument never assumed the values were positive, so a series that counts downward past zero sums correctly.
How is this different from adding a geometric sequence?
Here consecutive terms differ by a fixed amount, so pairing works. In a geometric sequence they are multiplied by a fixed ratio, the pairs no longer match, and a different formula built on cancellation is needed.