Sum of an AP
The sum of the first n terms of an arithmetic progression.
S_n = \dfrac{n}{2}\left[\,2a + (n - 1)d\,\right]
1What it means
Adds up the first n terms of an AP in one step. An equivalent form is Sₙ = n/2 (first term + last term).
2Variables
| Symbol | Meaning |
|---|---|
| S_n | Sum of the first n terms |
| n | Number of terms |
| a | First term |
| d | Common difference |
3Worked examples
Example 1 Worked solution
Q. Find the sum of the first 10 terms of 3, 7, 11, …
- a = 3, d = 4, n = 10.
- S₁₀ = 10/2 [2×3 + (10−1)×4].
- = 5 [6 + 36] = 5 × 42.
✓ S₁₀ = 210
4Where it's used
- ✦Totalling regular savings or repayments.
- ✦Summing scores, rows of seats, or stacked objects.
5Tips & common mistakes
- !If you know the last term, use Sₙ = n/2 (a + last term) — it is faster.