Have you heard about the concept of the binomial theorem, it is a kind of theorem that is used in order to calculate the values of some expanded versions of algebraic expression. The arrangement of the coefficients which are binomial in nature into a triangular shape or form is known as the pascal’s triangle. The applications of pascal’s triangle range from the theory of probability to algebra and combinatorics. It is used to find the chances or probability of a coin to be heads or tails. The numbers present in this triangle are arranged and managed in a way that every number is the addition of the other two numbers which are placed above it in the arrangement. In this article, we will try to cover some basic concepts regarding pascal’s triangle such as the pattern of pascal’s triangle, significant notes, and a detailed analysis about them.
The Pattern of Pascal’s Triangle
As mentioned above, a triangle that is used to arrange the binomial coefficients in a triangular shape or form is said to be a pascal’s triangle. The term was denoted after the name of a famous mathematician and physicist ‘Dr. Blaise Pascal’. The patterns that are found in the triangle of pascal are himself given by him. The following points analyses the patterns seen in this triangle.
- There are various rows present in the pascal’s triangle, these rows may have prime numbers (a number having two factors: itself and the number 1). If any of these rows have a prime number, then all the elements present in the row shall be divisible by that particular prime number. For example, A row of 1, 5, 10, 5, 10, and 1.
- There are various types of numbers that are situated in Pascal’s triangle such as the natural numbers, the pentalobe numbers, the tetrahedral numbers, the triangular number, etc. For example, the second diagonal row is full of triangular numbers, the third diagonal row is full of tetrahedral numbers.
- We can also get the Fibonacci series in the pascal’s triangle by adding different elements or numbers to the triangle. Fibonacci series is a series of numbers that forms a sequence where every second number is the sum of the previous ones.
- The value, when added in the ‘n’ roe, is equivalent to 2n. Let us take an example, in the fourth row that is 1, 4, 4, 6, 1. The addition of these elements gives us 1 + 4 + 4 + 6 + 1 which is equal to 16 = four times two.
Important notes to be Remembered about Pascal’s Triangle
In this paragraph, we will try to sum up every point that we studied about Pascal’s triangle in the previous few sections. Some of the points are as follows.
- The arrangement of the coefficients which are binomial in nature into a triangular shape or form is known as the pascal’s triangle. It was given after the name of Blaise Pascal.
- The element present in the triangle can be calculated by finding the sum of the adjoint points in the row which is proceeding.
If you want to learn about Pascal’s triangle in a detailed manner, in a fun way, and in an interactive manner, you may visit the website of Cuemath.
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