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CISCEICSE ICSE Class 6
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अभ्यासक्रम

Patterns in Whole/Natural Numbers

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Topics

  • Introduction
  • Sum of First n Odd Numbers
  • Sum of Natural Numbers
  • Sum of First n Even Numbers
  • Repeated 1's Pattern
  • Key Points Summary
CISCE: Class 6

Introduction

Patterns help us see order in numbers and predict what comes next. They're like mathematical magic tricks that work every time! Here are four important patterns that will amaze you.

CISCE: Class 6

Sum of First n Odd Numbers

Step-by-Step Understanding:

  • Start with 1 (that's 1² = 1)
  • Add the next odd number, 3: 1 + 3 = 4 (that's 2² = 4)
  • Add the next odd number, 5: 1 + 3 + 5 = 9 (that's 3² = 9)
  • Add the next odd number, 7: 1 + 3 + 5 + 7 = 16 (that's 4² = 16)

The sum of the first n odd numbers always equals n² (a perfect square).

CISCE: Class 6

Sum of Natural Numbers

Step-by-Step Understanding:

  1. Natural numbers: 1, 2, 3, 4, 5, ...
  2. Formula: 1 + 2 + 3 + ... + n = `"n(n + 1)"/ 2`
  3. Example: 1 + 2 + 3 + 4 + 5 = `"5 × (5 + 1)" / "2"` = 15

  • When Triangle 1 and Triangle 2 are combined, they form a rectangle of dots:
    Rows: 6 (the total number of natural numbers, n)
    Columns: 7 (each row sums to n+1)
  • So, the rectangle contains n(n+1) dots in total.
  • Since this rectangle is formed by two identical triangles, the number of dots in one triangle is
  • `"n(n + 1)"/ 2`
CISCE: Class 6

Sum of First n Even Numbers

Step-by-Step Understanding:

  1. Even numbers: 2, 4, 6, 8, 10, ...
  2. Formula: 2 + 4 + 6 + ... + 2n = n(n + 1)
  3. Example: 2 + 4 + 6 + 8 + 10 = 5 × 6 = 30
n First, n Even Numbers Sum Formula: n(n+1)
1 2 2 1 × 2 = 2
2 2 + 4 6 2 × 3 = 6
3 2 + 4 + 6 12 3 × 4 = 12
4 2 + 4 + 6 + 8 20 4 × 5 = 20
5 2 + 4 + 6 + 8 + 10 30 5 × 6 = 30
CISCE: Class 6

Repeated 1's Pattern

When you multiply numbers made of repeated 1's (like 1, 11, 111, etc.), the result follows a mirrored pattern.

  • For example:
    1 × 1 = 1
    11 × 11 = 121
    111 × 111 = 12321
    1111 × 1111 = 1234321, and so on.
CISCE: Class 6

Key Points Summary

  • Odd Numbers: 1 + 3 + 5 + ... + (2n−1) = n²

  • Natural Numbers: 1 + 2 + 3 + ... + n = n(n+1)/2

  • Even Numbers: 2 + 4 + 6 + ... + 2n = n(n+1)

  • Repeated 1's: Create palindromic patterns when multiplied

Test Yourself

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