What is the Pascal’s Triangle useful for?

Although it is known as Pascal’s Triangle, the author of this triangle is not Blaise Pascal. There are documents showing it was already known by the Chinese and Indian People a long time before the birth of Pascal. However, this triangle became famous after the studies made by this French philosopher and mathematician in 1647. One of the Pascal’s findings concerns the fact that `2^n` can calculate the addition of the elements of a line, having in mind that `n` is the number of the line. Other mathematicians have followed Pascal’s Works having found several other particular properties later on. Actually, Pascal’s Triangle is known by other names in other countries such as: “Tartaglia Triangle, “Yang Hui Triangle” or simply as “Arithmetic Triangle”.

How do I build this triangle?

This triangle is formed by numbers which are symmetrically placed according to a certain alignment. Each number of a certain line corresponds to the addition of the two numbers immediately adjacent to the previous line. You can see through the following animation how the first five lines of the Pascal’s Triangle are formed.

But... What is this triangle useful for?

Due to the way numbers are arranged, it is possible to find several properties among them. Those properties are useful in some mathematical calculations and they were used in ancient times to calculate the square or cubic roots, or more recently in the rule of probabilities. The terms of Newton binomial can be found from the numbers contained in the triangle, as well as, the terms which form Fibonacci sequence. In its diagonal we can also find the triangular numbers or the square numbers. Several patterns have been found recently such as the drawing of the Sierpinski triangle.

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