Patterns of Sequences

A sequence is a list of numbers arranged in a particular order, following a specific rule or pattern. Recognising these patterns helps us predict future terms and solve problems involving series and progressions.

 

Types of Sequences

1. Arithmetic Sequence (A.P.)

Each term increases (or decreases) by the same amount called the common difference (d).

$$T_n = a + (n - 1)d$$

• $a$ = first term
• $d$ = common difference
• $T_n$ = the $n$th term

Example: 3, 7, 11, 15, … has $d = 4$ and $a = 3$

 

2. Geometric Sequence (G.P.)

Each term is multiplied by a fixed number called the common ratio (r).

$$T_n = ar^{n-1}$$

• $a$ = first term
• $r$ = common ratio

Example: 2, 6, 18, 54, … has $r = 3$ and $a = 2$

 

3. Other Patterns

Not all sequences are arithmetic or geometric. Some follow square numbers, cube numbers, or even custom rules.

• Square number sequence: 1, 4, 9, 16, 25, …
• Fibonacci sequence: 1, 1, 2, 3, 5, 8, … (each term = sum of previous two)

 

Finding the Rule

To find the pattern rule for a sequence:

1. Check for a constant difference → Arithmetic
2. Check for a constant ratio → Geometric
3. Try recognising squares, cubes, or Fibonacci-style logic

Worked Example 1

Worked Example 2

Practice Problem