Patterns in Squares and Cubes
There are some interesting patterns in squares and cubes which often show some distinct properties and mathematical relations. Here are some important patterns which every student should know:
Patterns in Square Numbers
There are various patterns in the square numbers, some of those patterns are:
- Difference between square numbers
The difference between any two consecutive squares is always an odd number. For example, Two consecutive squares ‘4’ and ‘9’ are given. Their difference is 9-4= 5, which is an odd number.
- Sum of consecutive natural numbers
Whenever we square any odd number, the resultant will always be the sum of two consecutive natural numbers. Suppose we take the square of ‘3’ which is ‘9’. Here, ‘9’ is the result of the addition of two consecutive numbers ‘4’ and ‘5’.
- Product of two consecutive even or odd natural numbers
The product of two consecutive even numbers or consecutive odd numbers is also an important pattern of square numbers. For example, ’25’ is the product of odd numbers 5⨯5. Similarly, ’64’ is the product of two even numbers 8⨯8.
- Adding first n odd numbers
A square of any number is obtained by the sum of first ‘n’ odd numbers. Suppose, a number is given ’25’. Here, 25 is obtained by the addition of the first 5 odd numbers i.e. (1+3+5+7+9).
- Adding triangular numbers
Triangular numbers are the numbers obtained by adding the next natural number and it forms an equilateral triangle. The formula to find triangular numbers is:
T(n)= 1+2+3+4…….+n
Now, by adding these triangular numbers, square numbers can be generated easily. For example, the square number is ‘4’, which is the addition of the first triangular number to itself i.e. (1+3).
Patterns in Cube Numbers
Some commons patterns in cubes are:
- Adding consecutive odd numbers
By adding consecutive odd numbers, we can easily find the next cube numbers. For example, the cube of ‘1’ is 1. Now, add the next pair of consecutive odd numbers to find the next cube. Here, 3+5= 8 which is the cube of ‘2’. Similarly, to find the cube of ‘3’, add the next set of consecutive odd numbers 7+9+11= 27.
- Difference of Cubes of Two consecutive positive integers
The difference between the two consecutive positive integers can give a cubic number. For example, The difference between 23 – 13 = 7. This represents 23 = 8. Similarly, the difference between 33 – 23 = 19 which represents 33 = 27.
- Triangular Number Pattern
Similar to square numbers, we can also find cubic numbers by adding the triangular numbers. For example, the cube number is ‘23‘, which is the addition of the next triangular numbers i.e. (1+2=3).
Squares and Cubes
Squares and Cubes are mathematical operations involving numbers that are essential in various areas of mathematics. Square of a number is obtained by multiplying the number by itself where as a cube of a number is obtained by multiplying the number by itself twice.
In this article, we will learn what is Square and Cube Number. We will also learn about Perfect Squares and Cube and Square and Cube charts 1 to 100.
Table of Content
- What is Square Number?
- What is Cube Number?
- Perfect Square and Cubes
- Squares and Cubes from 1 to 50
- Chart of Squares and Cubes 1 to 100
Contact Us