Key takeaways
- aₙ = a + (n - 1)d, where a is the first term and d is the common difference.
- Use (n - 1), not n, since the first term already counts as position 1.
- d = any term minus the term right before it.
- A negative d means the sequence is decreasing.
In this guide
What is an arithmetic sequence?
In an arithmetic sequence, you add the same fixed number, called the common difference, to get from one term to the next. 3, 7, 11, 15 is arithmetic with a common difference of 4. The nth term is just the value sitting at position n in that list.
Why does the nth term formula matter?
Counting term by term works for small sequences, but it's slow and error-prone for large ones. If seats increase by a fixed number every row, or savings grow by a fixed deposit every month, this formula gives you any single value instantly, no matter how far out it is.
A theatre with 12 rows doesn't need you to count seats in rows 1 through 11 to know how many are in row 12. One formula, one calculation.
The formula
| Symbol | What it means | Example |
|---|---|---|
| aₙ | The term you want, at position n | 49 |
| a | The first term in the sequence | 5 |
| n | The position (term number) you want | 12 |
| d | The common difference between terms | 4 |
How to find the nth term step by step
- 1Identify the first term, a, and the common difference, d. Subtract any term from the one right after it to find d.
- 2Subtract 1 from the term number you want. This gives
(n - 1). - 3Multiply by the common difference.
(n - 1) × d. - 4Add the first term.
aₙ = a + (n - 1)d.
Enter your own first term, common difference, and position below to find any term instantly.
Try it yourself
Pre-filled with the example — change any value.
Common mistakes to avoid
- Getting the common difference backwards. d = second term minus first term, not the other way around.
- Mixing this up with a geometric sequence, where terms are multiplied, not added — use the geometric sequence calculator for that case.
- Assuming d must be a whole number. It can be a fraction or decimal, like 0.5 or -1.5.
Tips and tricks
- Working backwards from a term value to find n? Rearrange to
n = (aₙ - a) ÷ d + 1. - To add up every term instead of finding just one, use the arithmetic sequence sum calculator.
- Spot an arithmetic sequence by checking that consecutive differences are all equal.
Where you'll use it in real life
- Venues: rows of seats that each gain a fixed number of extra chairs.
- Savings: a balance that grows by the same fixed deposit every period.
- Construction: stacked layers, like bricks or steps, that add a constant amount each layer.
- Scheduling: events spaced by a fixed interval, like every 15 minutes.
Quick summary
To find any term of an arithmetic sequence, multiply the common difference by (n - 1) and add the first term. It's a shortcut that skips counting through every earlier term. Try the calculator above with your own sequence to see the working.
Worked example
A theatre's front row has 5 seats, and each row behind it has 4 more seats than the row in front. How many seats are in row 12?
- 01aₙ = a + (n - 1)d
- 02a12 = 5 + (12 - 1)(4)
- 03a12 = 5 + 44
- 04a12 = 49
These numbers are pre-filled in the calculator above.
Frequently asked questions
How do I find the nth term of an arithmetic sequence?
Use aₙ = a + (n - 1)d, where a is the first term, d is the common difference, and n is the position you want.
What is the common difference?
It's the fixed number added to each term to get the next one. You find it by subtracting any term from the term right after it.
Can the common difference be negative?
Yes — a negative d means the sequence decreases, like 20, 17, 14, 11, where d = -3.
What's the difference between arithmetic and geometric sequences?
Arithmetic sequences add a fixed amount each step. Geometric sequences multiply by a fixed ratio each step — see the geometric sequence calculator.
