Olympiad Notes: Patterns | Science Olympiad for Class 5 PDF Download

What is it and how is it useful?

• Mathematics is the study of patterns found everywhere in nature, our homes, and the universe.
• It helps us understand why these patterns exist and how they work.
• Patterns can be observed in daily activities like games, technology, and space.
• Mathematics aims to not only identify patterns but also explain them.
• These explanations can be applied to various fields, such as medicine (e.g., understanding genomes to diagnose and cure diseases).

Pattern in Numbers

• Number patterns are some of the most basic patterns in mathematics.
• These patterns often involve whole numbers like 0, 1, 2, 3, 4, etc.
• The branch of mathematics that studies these patterns is called number theory.
• Number sequences are a type of pattern studied in number theory.

Examples of Number Patterns:

• All 1's: 1, 1, 1, 1, 1, ...
• Counting Numbers: 1, 2, 3, 4, 5, 6, 7, ...
• Odd Numbers: 1, 3, 5, 7, 9, ...
• Even Numbers: 2, 4, 6, 8, 10, ...
• Triangular Numbers: 1, 3, 6, 10, 15, 21, ...
• Squares: 1, 4, 9, 16, 25, ...
• Cubes: 1, 8, 27, 64, 125, ...
• Virahanka Numbers: 1, 2, 3, 5, 8, 13, 21, ...
• Power of 2: 1, 2, 4, 8, 16, 32, 64, ...
• Power of 3: 1, 3, 9, 27, 81, 243, ...

Visualizing Number Sequence

• All 1's
  
• Counting Numbers
  
• Odd Numbers
  
• Even Numbers
  
• Triangular Numbers
  
• Squares
  
• Cubes
  

Relation among Number Sequence

Summing Odd Numbers Forms Square Numbers

• 1 = 1
• 1 + 3 = 4 
• 1  + 3 + 5  = 9
• 1 + 3 + 5 + 7 = 16
• 1  + 3 + 5 + 7 + 9 = 25

Adding Up and Down Sequence to form Squares

• 1 = 1
• 1 + 2 + 1 = 4
• 1 + 2 + 3 + 2 + 1 = 9
• 1 + 2 + 3 + 4 + 3 + 2 + 1 = 16
• 1 + 2 + 3 + 4 + 5 + 4 + 3 + 2 + 1 = 25

Patterns are not only found in numbers but also in shapes.

For Example:

• A sequence of growing squares.
• A sequence of triangles increasing in size.
  
  

Question for Olympiad Notes: Patterns

Try yourself:

Which number sequence is formed by adding consecutive odd numbers?

• A.

  All 1's
• B.

  Triangular Numbers
• C.

  Squares
• D.

  Odd Numbers

Explanation

- Triangular numbers are formed by adding consecutive natural numbers.
- For example, 1, 3, 6, 10, 15, etc., are triangular numbers.
- Adding consecutive odd numbers forms a sequence of triangular numbers.

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Shape Related to Number Sequence

• Shape sequences and number sequences can be related in surprising and helpful ways.
• Example: Shape Sequence of Regular Polygons . Regular Polygons as shapes with equal-length sides and equal angles.
• Triangle: (3 Equal Sides)
  
• Quadrilateral: (4 Equal Sides)
  
• Pentagon: (5 Equal Sides)
  
• Hexagon: (6 Equal Sides)