Key takeaways
- Sₙ = n/2 × [2a + (n - 1)d] adds up the first n terms at once.
- You only need the first term, the common difference, and how many terms — not the last term.
- n is how many terms you're adding, not the value of any term.
- If d is negative, the running total can grow more slowly or even shrink.
In this guide
What is the sum of an arithmetic sequence?
If you have a list of numbers that each increase by the same fixed amount (an arithmetic sequence), the sum is what you get by adding all of them together, from the first term up to the nth term.
Why does the sum formula matter?
Listing and adding every term by hand doesn't scale — nobody wants to manually add 50 growing numbers. This formula gets the exact total in one step, which is exactly what you need for totals like theatre capacity or combined savings.
Twelve rows that grow by 4 seats each, starting from 5, still add up to a specific, exact total — no counting chairs required.
The formula
| Symbol | What it means | Example |
|---|---|---|
| Sₙ | The total of the first n terms | 324 |
| n | How many terms you're adding | 12 |
| a | The first term | 5 |
| d | The common difference between terms | 4 |
How to find the sum step by step
- 1Identify a, d, and n. The first term, common difference, and number of terms to add.
- 2Calculate (n - 1)d and add it to 2a. This gives
2a + (n - 1)d. - 3Multiply the result by n.
- 4Divide by 2 to get Sₙ.
Enter your own a, d, and n below to see the running total worked out step by step.
Try it yourself
Pre-filled with the example — change any value.
Common mistakes to avoid
- Forgetting the factor of 2 on a. Inside the brackets it's
2a, not justa. - Confusing this with the nth term formula, which gives one term, not a running total — see the nth term calculator for that.
- Skipping the divide-by-2 step, which doubles your answer.
Tips and tricks
- To find a single term rather than a total, switch to the nth term calculator.
- Sanity-check small cases by hand: with n = 1, Sₙ should just equal a.
- For sequences that multiply instead of add, use the geometric series sum calculator.
Where you'll use it in real life
- Venues: total seating across rows that each add a fixed number of extra seats.
- Savings: total money saved after deposits that increase by a fixed amount each period.
- Fitness: total distance run across days where the daily distance increases by a fixed amount.
- Construction: total materials used across layers that each add a constant amount.
Quick summary
The arithmetic sum formula adds every term from the first to the nth without listing them all: multiply n by 2a plus (n-1)d, then divide by 2. It's exact and instant compared to adding term by term. Try the calculator above with your own sequence.
Worked example
The same theatre has 12 rows. Row 1 has 5 seats, and each row after it has 4 more seats than the one before. How many seats does the whole theatre have?
- 01Sₙ = n/2 × [2a + (n - 1)d]
- 02S12 = 12/2 × [2(5) + (12 - 1)(4)]
- 03S12 = 6 × [10 + 44]
- 04S12 = 6 × 54
- 05S12 = 324
These numbers are pre-filled in the calculator above.
Frequently asked questions
How do I find the sum of an arithmetic sequence?
Use Sₙ = n/2 × [2a + (n - 1)d], where a is the first term, d is the common difference, and n is the number of terms.
Do I need to know the last term to use this formula?
No — this version only needs the first term, the common difference, and how many terms you're adding. It works out the last term internally.
What if the common difference is negative?
The formula still works. A negative d means later terms are smaller, so the sum grows more slowly (or even decreases) as n increases.
Is this the same as just adding every term one by one?
It gives the exact same answer, but instantly — no need to list and add dozens or hundreds of terms by hand.
