Arithmetic Series Calculator (Sum of AP)
This arithmetic series calculator — the sum of AP calculator, if that is the phrase your textbook uses — adds up the first n terms of an arithmetic sequence with steps. Give the first term and how many terms, plus either the common difference or the last term, and both standard formulas are worked out and compared.
Take 2, 5, 8, 11, … for ten terms. The last is 29, and pairing the ends gives 2 + 29 = 31, then 5 + 26 = 31, then 8 + 23 = 31 — five pairs of 31, so the total is 155. That pairing is not a trick; it is the formula itself, and this page builds it on whatever numbers you type in.
Sum of an AP, or Arithmetic Series — the Same Thing
Two names cause more confusion here than any of the mathematics. A sequence is the list: 2, 5, 8, 11. A series is what you get when you add that list up: 2 + 5 + 8 + 11. So the sum of an AP and an arithmetic series are one and the same, and the total is written Sₙ.
Which words you meet depends on where you studied. Indian textbooks and the NCERT syllabus speak of an arithmetic progression, shortened to AP, and ask for the sum of the first n terms of an AP. British and American books more often say arithmetic series. There is no difference in the formula, the working, or the answer — so if you arrived here searching for either phrase, you are in the right place.
If what you actually need is a single term rather than a total — the 20th term, say, rather than everything up to it — that is a different question, and the arithmetic sequence calculator answers it directly.
Where the Formula Comes From
Write the list out and fold it in half. Pair the first term with the last, the second with the second-last, and keep going. Something neat happens: every pair adds to the same total, because whatever one end gains as you step inwards, the other end loses.
For 2, 5, 8, …, 29 the pairs are (2 + 29), (5 + 26), (8 + 23), (11 + 20), (14 + 17). All five come to 31. Ten terms make five pairs, so the sum is 5 × 31 = 155.
Write that as a rule and the formula appears: n terms make n/2 pairs, and each pair totals a₁ + aₙ. Hence:
Sₙ = n/2 × (a₁ + aₙ)
The famous version of this is the schoolboy Gauss adding 1 to 100 in seconds: fifty pairs, each adding to 101, so 50 × 101 = 5050. This calculator does the same fold on your own numbers rather than telling you the story, and prints the pairs so you can see the totals match.
There is a second way to read the same line, and it is the easiest one to keep in your head: a sum is a count times an average. The average of the two ends is (a₁ + aₙ)/2, and because the terms are evenly spaced that is the average of the whole list. For our example the average is 15.5, and ten terms averaging 15.5 come to 155.
The Two Formulas, and When to Use Each
Textbooks print two versions, which makes them look like two rules to memorise. They are not.
Sₙ = n/2 (a₁ + aₙ) — use this when the question gives you the last term.
Sₙ = n/2 (2a₁ + (n − 1)d) — use this when it gives you the common difference instead.
Put aₙ = a₁ + (n − 1)d into the first and the second falls straight out, so they are one formula wearing two coats. Whichever you type in, this page works out both and shows them landing on the same number — and refuses to display anything if they ever disagree.
A worked case with a negative difference: 10, 8, 6, 4, 2, 0 has a₁ = 10, d = −2 and n = 6. The first form needs the last term, which is 0, giving 6/2 × (10 + 0) = 30. The second needs no last term: 6/2 × (20 + 5 × −2) = 3 × 10 = 30. Same answer, different route.
The Odd Case: When One Term Has No Partner
The pairing story has a gap in it that most explanations quietly skip, and it is exactly the point where a careful student objects. If the number of terms is odd, the folding leaves one term stranded in the middle with nobody to pair with.
Take nine terms of 2, 5, 8, … The pairs are (2 + 26), (5 + 23), (8 + 20) and (11 + 17) — four pairs of 28 — and 14 sits alone in the middle. Four pairs give 112, and adding the lonely 14 gives 126.
Now look at that middle term again. It is 14, and the average of the two ends is (2 + 26)/2 = 14 as well. The stranded term is exactly halfway between the ends, so counting it as half a pair is not a fudge — it is arithmetically exact, which is why n/2 still works when n is odd and the formula needs no special case. This calculator marks that term and says so rather than leaving you to wonder.
How to Use This Arithmetic Series Calculator
The screen has three parts: the answer band at the top, four boxes below it, and the keypad underneath.
1. Give the first term and how many. a₁ and n are always needed — a sum has to know where it starts and how far it goes. n must be a whole number of 1 or more.
2. Then give either d or the last term. Whichever your question hands you. The box you leave empty turns dashed with a ?, and once you press = it turns green and shows the value that was worked out, marked ↑ worked out. Fill in both and they are checked against each other instead.
3. Press = and open the working. You get the derived value first, then your own terms paired up with each total, then the count multiplied out, then both formulas side by side, and finally the terms added one at a time with a running total.
Every key and control, in one place:
| first term a₁ | Where the series starts. Always needed |
| common difference d | What is added each step; negative for a falling series |
| how many terms n | How far the sum goes. A whole number, 1 or more |
| last term aₙ | The final term. Give this or d, not necessarily both |
| ← → ↑ ↓ | Move between the four boxes |
| ± | Flips the sign of the box you are in |
| ⁄ | The fraction bar. 3⁄4 stays a fraction all the way through |
| . | Decimal point. 0.25 is read as one quarter, exactly |
| ⌫ | Deletes one character from the box you are in |
| AC | Clears all four boxes |
| = | Adds them up |
| My keyboard | Raises your own keyboard instead of the pad, for that box |
| copy | The icon in the band copies the sum |
Nothing is rounded at any stage, so a common difference of one third stays one third rather than becoming 0.333 — which matters, because a rounding error repeated across a hundred terms stops being invisible. There is no cap on how many terms you can add, either: a million is answered as readily as ten.
Arithmetic Series FAQ
Is the sum of an AP the same as an arithmetic series?
What is the formula for the sum of an arithmetic series?
Why does the sum formula have n divided by 2 in it?
What happens when the number of terms is odd?
What is the sum of the first 100 natural numbers?
Can the sum of an arithmetic series be negative or zero?
Fold the list, pair the ends, count the pairs. Whether your book calls it the sum of an AP or an arithmetic series, that one picture is the whole formula — and it is why the answer is a count times an average rather than anything you need to memorise.