Case 1
- First term a₁
- 2
- Common difference d
- 3
- Term position n
- 10
29.0000
Open with these valuesArithmetic Sequence Calculator
29.0000
Result: 29.0000Each term sits a fixed step d away from the one before it, so the nth term is a + (n − 1) × d. Counting starts at n = 1, and the first term needs no step at all — that is where the minus one comes from. With a = 2 and d = 3 the tenth term is 2 + 27 = 29.
The numbers at a glance
Held fixed: First term a₁ 2.0000, Common difference d 3.0000.
| Term position n | Result |
|---|---|
| 3 | 8.0000 |
| 5 | 14.0000 |
| 8 | 23.0000 |
| 10Your value | 29.0000 |
| 13 | 38.0000 |
| 15 | 44.0000 |
| 18 | 53.0000 |
| 20 | 59.0000 |
Worked examples
Case 2
- First term a₁
- 1
- Common difference d
- 1
- Term position n
- 100
100.0000
Open with these valuesCase 3
- First term a₁
- 5
- Common difference d
- -2
- Term position n
- 6
-5.0000
Open with these valuesHow it's calculated
aₙ = a + (n − 1) × d
- StepEnter the first term — the value the sequence starts at.
- StepEnter the common difference: the fixed amount added at every step.
- StepEnter the position you want; n = 1 is the first term itself.
- ResultRead the nth term. The table shows the running sum beside it.
Reference table
| a, d, n | Sum of the first n terms | nth term |
|---|---|---|
| 0, 5, 1 | 0 | 0 |
| 10, 0, 4 | 40 | 10 |
| 2, 3, 10 | 155 | 29 |
| 5, -2, 6 | 0 | -5 |
| 1, 1, 100 | 5050 | 100 |
Questions
What is an arithmetic sequence?
An arithmetic sequence is a list of numbers where each term comes from adding a fixed amount — the common difference d — to the previous one: a, a+d, a+2d, … For example 2, 5, 8, 11, … has first term 2 and common difference 3. The gap between any two neighbours is always the same.
How do I find the nth term?
Use aₙ = a + (n − 1) × d, where a is the first term, d is the common difference and n is the position you want. For 2, 5, 8, … the tenth term is 2 + 9 × 3 = 29. You subtract 1 because the first term needs zero steps of d.
Does counting start at n = 0 or n = 1?
Here it starts at n = 1, so n = 1 returns the first term unchanged. Textbooks that index from zero write the same sequence as a₀ + k·d, and their k is one less than this n. Getting the two mixed up puts you exactly one term off.
What happens with a negative or zero common difference?
A positive d makes the sequence grow (2, 5, 8, …), a negative d makes it shrink (5, 3, 1, −1, …) and d = 0 keeps every term equal to the first. The same formula covers all three cases.
How is an arithmetic sequence different from a geometric one?
An arithmetic sequence adds a constant difference each step, so the differences are equal (2, 5, 8, 11). A geometric sequence multiplies by a constant ratio each step, so the ratios are equal (2, 6, 18, 54). Use this calculator for the additive kind.
How do I get the sum of the first n terms?
The sum is n ÷ 2 × (2a + (n − 1) × d), which is just n times the average of the first and last term. For a = 2, d = 3, n = 10 that gives 5 × 31 = 155. The middle column of the table above carries this sum for every row.
Sources and last check
Information, not professional advice.