a) Adding Triangular Numbers
A triangular number is a number that forms a triangle when represented as dots.
Example of triangular numbers:
1,3,6,10,15,21,…
Pattern:
Adding two consecutive triangular numbers results in a square number.
b) Numbers Between Two Square Numbers
Between two consecutive square numbers, the number of non-square numbers follows a pattern.
For example:
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Between 1² = 1 and 2² = 4, there are 2 non-square numbers (2, 3).
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Between 2² = 4 and 3² = 9, there are 4 non-square numbers (5, 6, 7, 8).
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Between 3² = 9 and 4² = 16, there are 6 non-square numbers (10, 11, 12, 13, 14, 15).
General Rule:
The number of non-square numbers between n² and (n+1)² is:
2n
For example, between 6² = 36 and 7² = 49:
2(6) = 12 non-square numbers
(c) Sum of Consecutive Odd Numbers
Pattern: The sum of the first n odd numbers gives the nᵀ square number.
1 = 1²
1 + 3 = 2² = 4
1 + 3 + 5 = 3² = 9
1 + 3 + 5 + 7 = 4² = 16
Thus, any perfect square is the sum of consecutive odd numbers starting from 1
(d) Expressing Square Numbers as Sum of Two Consecutive Numbers
Odd square numbers can be expressed as the sum of two consecutive numbers.
Examples:
3² = 9 = 4 + 5
5² = 25 = 12 + 13
7² = 49 = 24 + 25
General Formula:
n² = (n² – 1)/2 + (n² + 1)/2
(e) Product of Two Consecutive Odd or Even Numbers
Multiplying two consecutive odd or even numbers follows a special pattern:
(a – 1) × (a + 1) = a² – 1
Examples:
11 × 13 = (12 – 1)(12 + 1) = 12² – 1 = 143
29 × 31 = (30 – 1)(30 + 1) = 30² – 1 = 899
This pattern works for both odd and even numbers.
(f) Unique Patterns in Square Numbers
Pattern in Squares of 1, 11, 111, …
1² = 1
11² = 121
111² = 12321
1111² = 1234321
Pattern in Squares of Numbers with Repeated 6s
7² = 49
67² = 4489
667² = 444889
6667² = 44448889
Conclusion
Square numbers exhibit fascinating patterns that help in problem-solving and mathematical exploration. Recognizing these patterns makes learning squares more engaging and useful in real-life applications.