A sequence is a list of numbers arranged in a particular order according to a rule.
The two most common types are
1. Arithmetic Sequence: Sequence of numbers in which the difference between any two consecutive terms is always the same. This fixed difference is called the common difference (d). Each new term is obtained by adding or subtracting the same number from the previous term.
General Form: a, a + d, a + 2d, a + 3d, ...
2. Geometric Sequence: Sequence of numbers in which the ratio between any two consecutive terms is always the same. This fixed ratio is called the common ratio (r). Each new term is obtained by multiplying or dividing the previous term by the same number.
General Form: a, ar, ar², ar³, ar⁴, ...
The table below compares their main features.
| Feature | Arithmetic Sequence (AS) | Geometric Sequence (GS) |
|---|---|---|
| Rule | Add or subtract the same number each time. | Multiply or divide by the same number each time. |
| Common Element | Common Difference (d) | Common Ratio (r) |
| Relation Between Terms | The difference between consecutive terms is constant. | The ratio between consecutive terms is constant. |
| Operation Used | Addition or Subtraction | Multiplication or Division |
| Growth Pattern | Linear growth | Exponential growth |
| General Form | a, a + d, a + 2d, a + 3d, ... | a, ar, ar², ar³, ar⁴, ... |
| nᵗʰ Term | aₙ = a + (n − 1)d | aₙ = arⁿ⁻¹ |
| Sum of First n Terms | Sₙ = n/2 [2a + (n − 1)d] | Sₙ = a(rⁿ − 1)/(r − 1), r ≠ 1 |
| Example | 5, 11, 17, 23, 29, ... | 2, 6, 18, 54, 162, ... |