Chapter 1: Patterns and Sequences

1.1 Patterns

A pattern is a list of numbers or geometrical shapes arranged according to a specific rule or design.

Common Types of Number Patterns

• Odd Numbers: $1, 3, 5, 7, 9, \dots$ (pattern: adding $2$ to the preceding number).
• Even Numbers: $2, 4, 6, 8, 10, \dots$ (pattern: adding $2$ to the preceding number).
• Pascal's Triangle: A triangular array of numbers where each number inside is the sum of the two directly above it.
  • Row 1: $1$
  • Row 2: $1, 1$
  • Row 3: $1, 2, 1$
  • Row 4: $1, 3, 3, 1$
  • Row 5: $1, 4, 6, 4, 1$
• Fibonacci Numbers: A sequence of numbers where each term is generated by adding the two preceding terms.

  Formula: $T_n = T_{n-1} + T_{n-2}$ for $n \ge 3$

  Sequence: $0, 1, 1, 2, 3, 5, 8, 13, 21, 34, \dots$

Describing a Pattern

A pattern can be described using three main methods:

1. Using Words: Describe how each term is obtained from the previous term (e.g., "Add $3$ to the previous number").
2. Using Numbers: Expressed directly as numerical operations (e.g., $+3, +3, +3, \dots$).
3. Using Algebraic Expressions: Expressed as a mathematical expression involving $n$ (e.g., $2n + 1$, where $n = 1, 2, 3, \dots$).

1.2 Sequences

A sequence is an ordered set of numbers or objects that follows a specific pattern or rule.

Notation of Terms in a Sequence

• Each number in a sequence is called a term.
• The $n$-th term is denoted as $T_n$, where $n = 1, 2, 3, 4, \dots$
  • $T_1$ = $1^{\text{st}}$ term
  • $T_2$ = $2^{\text{nd}}$ term
  • $T_3$ = $3^{\text{rd}}$ term
  • $T_n$ = $n^{\text{th}}$ term (General Term)

1.3 Patterns and Sequences

General Term ($T_n$)

The general term $T_n$ allows us to find any term in a sequence without listing all the preceding numbers.

For an arithmetic sequence with a constant difference $d$ between consecutive terms:

$$T_n = T_1 + (n - 1)d$$

Where:

• $T_n$ = $n^{\text{th}}$ term
• $T_1$ = First term of the sequence
• $d$ = Common difference ($T_2 - T_1$)
• $n$ = Term position ($n = 1, 2, 3, \dots$)

Solving Problems Involving Sequences

To solve problems involving patterns and sequences:

1. Identify the pattern by calculating differences or ratios between consecutive terms.
2. Formulate the algebraic expression for $T_n$.
3. Substitute the desired position $n$ into $T_n$ to calculate the value of the term.