How to do Number Patterns
Let’s look at another pattern. Say we have a triangle.
Flip the triangle downwards and complete this image to form a complete triangle as given below:
The pattern is still a triangle, but the number of smaller triangles increases to 4. This bigger triangle now has two rows. What happens when we increase the number of rows and fill in the gaps with smaller triangles to complete that big triangle?
As we proceed to the 3rd row, how many smaller triangles do we have in total now? Now, this triangle has 9 smaller triangles.
What is the pattern here?
When,
r = 1, total no of triangles, ‘n’ = 1
When,
r = 2, n = 4
And when,
r = 3, n = 9
This can be summarised as:
| Number of rows | 1 | 2 | 3 |
| Number of smaller triangles | 1 | 4 | 9 |
You can see that as the size of the triangle increases, the number of smaller triangles also increases. This means that n is just equal to r², which is 1², 2², 3² = 1, 4, 9…
So how do we represent this triangular pattern algebraically? r²! That’s it! We have just defined patterns algebraically. Also, the above pattern can be defined as the growth pattern in algebra.
We can write the sequence of odd numbers like 1, 3, 5, 7, 9……………………….. (2n – 1). If we substitute ‘n’ values in the expression beginning from one for odd numbers, i.e. 2n – 1, we can easily calculate the nth odd number. Substitute n= 11 to find the 11th odd number, and the result is 21, i.e. 11th odd number. Similarly, the 100th odd number is 199. Thus, the algebraic expression representing a set of odd numbers is 2n -1. It must be noted that n is a set of whole numbers in this case.
Similar to patterns in numbers, we can figure out the pattern in figures. For example, consider the following figures:
In the figure given above, the first image is a pentagon with five sides. In the following figure, two pentagons are joined end to end, and the total number of sides is 9; in the next figure, the total number of sides is 13. It is observed that every other figure has 4 extra sides as compared to the previous one. Thus, the algebraic pattern that would define this sequence exactly is ‘4n + 1’, where n is any natural number. Therefore, if we substitute n = 3, we get the number of sides equal to 13. The 10th pattern in this sequence will have sides equal to 4 × 10 + 1, i.e. 41 sides.