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.

## Fibonacci Sequence

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.

## Sierpiński Triangle:

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.

## Binomial Expansion:

A binomial expression is the sum (or difference) of two terms. Binomial expansion is when this expression is raised to a power. For example:

**(x+1) ^{3}**

The result of this expansion is:

**x ^{3} + 3x^{2} +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:

**(2x+1) ^{4}**

This time the solution is:

**16x ^{4} + 32x^{3} + 24x^{2} + 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})(1^{2}) + 4(2x)(1^{3}) + 1(1^{4})

Expanding the brackets out, we will arrive at the solution above. Remember that (2x)^{4} will actually be 16x^{4 }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:

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:

## More patterns:

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:

**Catalan Numbers**

**Square Numbers (and other polygonal numbers)**

**Powers of 11**