Lecture Notes: Form Two Chapter One - Patterns and Sequences

Overview

This chapter covers three main topics:

• 1.1 Patterns
• 1.2 Sequences
• 1.3 Patterns and Sequences

1.1 Patterns

• Definition:

  • Patterns are lists of numbers or objects arranged based on a rule or design.

• Recognizing Number Patterns

  • Example Problem: Identify the next object or number in a sequence.
  • Solutions:
    • Add two dots to the previous object.
    • Add a triangle to the previous object.

• Exercises and Solutions:

  • Add 6 to the previous number.
  • Subtract 10 from the previous number.
  • Multiply the previous number by three.
  • Divide the previous number by three.
  • Add half to the previous number.
  • Subtract 0.3 from the previous number.

• Even and Odd Numbers

  • Example: Given a series of numbers (7, 12, 17, 22, 27, 67), identify odd and even number patterns.
  • Solution:
    • Odd numbers: 7, 17, 27, 37, 47, 57, 67 (Pattern: Add 10 to the previous number)
    • Even numbers: 12, 22, 32, 42, 52, 62 (Pattern: Add 10 to the previous number)

• Pascal's Triangle

  • Construction: Start with 1 at the top, each number is the sum of the two numbers directly above it.
  • Patterns:
    • Sequence of 1, 2, 3, 4, 5, 6 (Pattern: Add 1)
    • Sequence of 1, 3, 6, 10, 15 (Pattern: Add 2, 3, 4, 5)

• Fibonacci Numbers

  • Definition: Pattern starting with 0, 1, 1, each subsequent number is the sum of the previous two.

1.2 Sequences

• Definition:

  • A sequence is a set of numbers or objects arranged according to a certain pattern.

• Example Problems:

  • Determine if a set is a sequence based on a pattern.
    • If there is a consistent pattern (e.g., add 4), it is a sequence.
    • If there is no pattern, it is not a sequence.

• Completing Number Sequences:

  • Subtract 4, multiply by 3, subtract 8, divide by 5.

• Describing Patterns using Numbers, Words, and Algebraic Expressions:

  • Example: Sequence 1, 9, 17, 25, 33
    • Numbers: Plus eight
    • Words: Add eight to the previous number
    • Algebraic Expression: $1 + 8n$ where $n = 0, 1, 2, 3$

1.3 Patterns and Sequences

• Terms of the Sequence

  • $T_n$ represents the $n^{th}$ term where $T$ is the term and $n$ is the position.

• Example Problems:

  • Example 1: Given sequence 4, 8, 12, 16 - Identify terms.
    • $T_1$ is 4, $T_2$ is 8, $T_3$ is 12, $T_4$ is 16.
  • Example 2: Identify fifth term in a sequence.
    • Pattern: Add 8 to the previous number
    • First five terms: $T_1$ is 2, $T_2$ is 10, $T_3$ is 18, $T_4$ is 26, $T_5$ is 34
  • Example 3: Determine which term is 40 in the sequence 65, 60, 55, 50.
    • Pattern: Subtract 5 from the previous number
    • List of terms: $T_1$ is 65, $T_2$ is 60, $T_3$ is 55, $T_4$ is 50, $T_5$ is 45, $T_6$ is 40

Conclusion

• For further questions or clarifications, viewers are encouraged to comment below.