Arithmetic Sequences - Common Difference and Sum
An arithmetic sequence is a list of numbers in which each term is obtained from the previous one by adding (or subtracting) a fixed amount. That fixed amount is called the common difference, usually written d. Arithmetic sequences are the most fundamental type of sequence in mathematics and appear constantly in real life.
Key Terms
| Term | Symbol | Meaning |
|---|---|---|
| First term | a | The starting value of the sequence |
| Common difference | d | The fixed amount added each time (can be negative) |
| nth term | an | The value of the term in position n |
| Number of terms | n | How many terms are in the sequence |
The nth Term Formula
an = a + (n − 1)d
Where a = first term, d = common difference, n = position of the term.
Sum of an Arithmetic Sequence
Sn = n/2 × (first term + last term)
or equivalently: Sn = n/2 × (2a + (n−1)d)
This formula was famously used by the young Gauss to add the numbers 1 to 100 in seconds: S = 100/2 × (1 + 100) = 50 × 101 = 5,050.
Worked Examples
a = 7, d = 4. a20 = 7 + (20−1) × 4 = 7 + 76 = 83.
From 5th to 9th is 4 steps: 38 − 22 = 16, so d = 16 ÷ 4 = 4.
a5 = a + 4d → 22 = a + 16 → a = 6.
Sequence: 6, 10, 14, 18, 22, 26, 30, 34, 38 … ✓
a = 3, d = 5, n = 15. S15 = 15/2 × (2(3) + 14(5)) = 15/2 × (6 + 70) = 15/2 × 76 = 570.
a = 15, d = 2, n = 20. Last row: a20 = 15 + 19(2) = 53.
Total = 20/2 × (15 + 53) = 10 × 68 = 680 seats.
Key Takeaways
- Arithmetic sequence: constant difference d between consecutive terms.
- nth term: an = a + (n−1)d.
- Sum of n terms: Sn = n/2 × (first + last) = n/2 × (2a + (n−1)d).
- d positive → increasing sequence; d negative → decreasing sequence.
Practice Questions
- Find the 12th term of: 5, 9, 13, 17, …
- The 4th term of an arithmetic sequence is 17 and d = 3. Find the first term and the 10th term.
- Find the sum of the integers from 1 to 50.
- Find the sum of the first 20 terms of: 2, 6, 10, 14, …
- How many terms are in the sequence 7, 14, 21, …, 105? Find their sum.