If Natural Number is Square Number, then it has to be the sum of Successive Odd Numbers starting from 1

Property 1	Property 2	Property 3	Property 4	Property 5
Property 6	Property 7	Property 8

Before you understand this property, you are adviced to read: What are Natural Numbers ? What are Square Numbers ? What are Odd Numbers ? Observe the following table 1: Perfect Squares Sum of Odd Numbers 4 1 + 3 9 1 + 3 + 5 16 1 + 3 + 5 + 7 25 1 + 3 + 5 + 7 + 9 36 1 + 3 + 5 + 7 + 9 + 11 49 1 + 3 + 5 + 7 + 9 + 11 + 13 64 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 81 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 100 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 In the above table 1: Column 1 represents Square numbers or perfect squares Column 2 represents Sum of odd numbers starting from 1 Now, you can observe that in each row sum of odd numbers is equal to its corresponding square number. That is: 4 = 22 = sum of 1st two odd numbers 9 = 32 = sum of 1st three odd numbers 16 = 42 = sum of 1st four odd numbers 25 = 52 = sum of 1st five odd numbers 36 = 62 = sum of 1st six odd numbers 49 = 72 = sum of 1st seven odd numbers 64 = 82 = sum of 1st eight odd numbers 81 = 92 = sum of 1st nine odd numbers 100 = 102 = sum of 1st ten odd numbers This explains the property of square numbers which says that: If a Natural Number is a square number, then it has to be the sum of successive odd numbers starting from 1 Or we can say that: "Square number is the sum of successive odd numbers starting from 1" With this property we also get that: If a Natural Number cannot be expressed as sum of successive odd numbers starting from 1, then the Natural Number is also not a perfect square.