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Some More Interesting Patterns

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description

  • Adding triangular numbers
  • Numbers between square numbers
  • Adding odd numbers
  • Sum of consecutive natural numbers
  • Product of two consecutive even or odd natural numbers
  • Some more patterns in square numbers

notes

1. Adding triangular numbers : 
The triangular number means numbers whose dot patterns can be arranged as triangles. 

If we combine two consecutive triangular numbers , we get a square numbers. 

2. Numbers between square numbers :
Let us now see if we can find some interesting pattern between two consecutive square numbers.

Between 12(=1) and 22(= 4) there are two (i.e., 2 × 1) non square numbers 2, 3. 
We have `4^2 = 16`     and  ` 5^2 = 25`
Therefore, `5^2 – 4^2 = 9` 
Between `16(= 4^2)` and `25(= 5^2)` the numbers are 17, 18, ... , 24 that is, eight non square numbers which is 1 less than the difference of two squares. 
Consider `7^2` and `6^2`. Can you say how many numbers are there between `6^2` and `7^2`? 

If we think of any natural number n and (n + 1), then, 
`(n + 1)^2 – n^2 = (n^2 + 2n + 1) – n^2 = 2n + 1`. 
We find that between `n^2` and `(n + 1)^2` there are 2n numbers which is 1 less than the difference of two squares.
Thus, in general we can say that there are 2n non perfect square numbers between the squares of the numbers n and (n + 1).

3. Adding odd numbers 
Consider the following 
1  [one odd number] = 1 = `1^2` 
1 + 3  [sum of first two odd numbers] = 4 = `2^2` 
1 + 3 + 5  [sum of first three odd numbers] = 9 = `3^2` 
1 + 3 + 5 + 7  [... ] = 16 = `4^2` 
1 + 3 + 5 + 7 + 9  [... ] = 25 = `5^2` 
1 + 3 + 5 + 7 + 9 + 11  [... ] = 36 =`6^2` 
So we can say that the sum of first n odd natural numbers is `n^2`. Looking at it in a different way, we can say: ‘If the number is a square number, it has to be the sum of successive odd numbers starting from 1. 
If a natural number cannot be expressed as a sum of successive odd natural numbers starting with 1, then it is not a perfect square. 

4. A sum of consecutive natural numbers 
Consider the following 

`9^2 = 81 = 40 + 41` 
`11^2 = 121 = 60 + 61` 
`15^2 = 225 = 112 + 113`
We can express the square of any odd number as the sum of two consecutive positive integers.

5. Product of two consecutive even or odd natural numbers 
11 × 13 = 143 = `12^2` – 1 
Also 11 × 13 = (12 – 1) × (12 + 1) 
Therefore, 11 × 13 = (12 – 1) × (12 + 1) = `12^2` – 1 
Similarly, 13 × 15 = (14 – 1) × (14 + 1) = `14^2` – 1 
29 × 31 = (30 – 1) × (30 + 1) = `30^2` – 1 
44 × 46 = (45 – 1) × (45 + 1) = `45^2` – 1 
So in general we can say that (a + 1) × (a – 1) = `a^2` – 1.

6. Some more patterns in square numbers
Observe the squares of numbers; 1, 11, 111 ... etc. They give a beautiful pattern: 

Another interesting pattern . 

 

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