Pascal’s triangle is a triangle which follows one simple rule, each number is made by adding the two numbers above it together. Named after Blaise Pascal, a mathematician who did a lot of work studying the triangle, it was actually created hundreds of years before.
Although it seems a very simple arrangement of numbers, it contains many hidden patterns including Sierpinski’s triangle, the powers of two and the numbers associated with binomial expansion.
By finding the sums of shallow diagonals (shown above) the Fibonacci sequence can be found in Pascal’s triangle. This pattern is a sequence where each term is found by adding the previous two terms together.
The Sierpiński triangle is an example of a fractal, a patter which continues repeating itself infinitely as you zoom in closer. It can be found within Pascal’s triangle by colouring in all of the odd numbers:
Powers of 2:
The sum of each row in Pascal’s triangle is a power of two:
This is because each number on a row will be added twice to form the row below, therefore doubling the sum of the row.
A binomial expression is the sum (or difference) of two terms. Binomial expansion is when this expression is raised to a power. For example:
The result of this expansion is:
x3 + 3x2 +3x + 1
The coefficients in the answer are the same as the fourth row of pascals triangle. This is always the case for the expansion of x+1. When the expression is not x+1, these numbers are still significant. For example:
This time the solution is:
16x4 + 32x3 + 24x2 + 8x + 1
Firstly, we look at the fifth row of pascals triangle, which is: 1, 4, 6, 4, 1. Then, the terms are written out. The x terms are written in decreasing powers, starting at four. The constant, in the case 1, is written is ascending powers.
1((2x)4) + 4((2x)3)(1) + 6((2x)2)(12) + 4(2x)(13) + 1(14)
Expanding the brackets out, we will arrive at the solution above. Remember that (2x)4 will actually be 16x4 because both the 2 and x are raised to the power of 4.
Binomial expansion can also be found using factorials, which is much easier when the powers are much higher.
Primes and Multiples:
If the first number in a row (excluding the 1) is a prime, all other numbers in the row will be divisible by it. For the proof of this, go to: http://mathforum.org/mathimages/index.php/Pascal’s_triangle#Patterns_about_Primes
Triangular numbers are found by finding the sum of consecutive natural numbers, they also form triangles when drawn out:
The nth term of this sequence is 1/2 n(n+1). This can be shown by forming a parallelogram with two of the triangles, with side lengths of n and n+1. The area of the parallelogram is n(n+1). Halving this for one triangle produces the formula given above.
The triangular numbers can also be found in pascal’s triangle. They form the third column:
There are a great number of patterns in Pascal’s triangle, and I have listed some of the most interesting ones here. However there are many more patterns such as:
Square Numbers (and other polygonal numbers)
Powers of 11