nth Term of an AP
The value of the nth term of an arithmetic progression.
a_n = a + (n - 1)d
1What it means
In an arithmetic progression (AP) each term increases by a fixed common difference d. This formula jumps straight to any term without listing all the ones before it.
2Variables
| Symbol | Meaning |
|---|---|
| a_n | The nth term |
| a | First term |
| n | Term number |
| d | Common difference (term − previous term) |
3Worked examples
Example 1 Worked solution
Q. Find the 10th term of 3, 7, 11, 15, …
- First term a = 3, common difference d = 7 − 3 = 4.
- a₁₀ = a + (10 − 1)d = 3 + 9 × 4.
- = 3 + 36.
✓ a₁₀ = 39
4Where it's used
- ✦Quickly finding a far-off term in a sequence.
- ✦Salary/instalment problems that grow by a fixed amount each period.
5Tips & common mistakes
- !d can be negative if the sequence decreases.
- !It is (n − 1)d, not nd — the first term already counts as one.