Pascal’s triangle is an arithmetic triangle. It can raise a binomial to any power. The triangle consists of the coefficients of monomials comprising the formula of the n-th power of the sums of two numbers.

Pascal’s triangle is an isosceles triangle, which has one at the apex and a pair of ones on the sides.  Each number is equal to the sum of the two numbers above it. The rows of Pascal’s triangle are symmetrical with respect to the vertical axis.

The triangle can be continued infinitely.

Pascal's triangle

where n is a natural number and:

Pascal's triangle

Construction of Pascal’s triangle

 

We will write down the numbers starting from n=1 and ending with n=5 in order to note the regularity in the formula for the n-th power of a binomial at various values of n.

Construction of Pascal's triangle

Considering the formulae it can be observed that the right-hand part of each of them comprises a polynomial containing n+1 members where n is the index of power of the binomial.

The first member of the polynomial is equal to an, i.e. it equals the product of an and b0. Then moving to each subsequent member the index of power of a decreases by 1 and the index of power of b increases by 1, i.e. the sum of the indexes of power in each summand is equal to n.

The situation with coefficients is harder. In order to identify the pattern, we write, in the correct order in a row, the coefficients of polynomials at n=2 and then at n=3.

Construction of Pascal's triangle

The first and the last coefficients in the second row are equal to 1. It can readily be noted that the second coefficient can be derived by summing up adding the numbers 1 and 2 written above it, the third coefficient – by adding the numbers 2 and 1 written above it.

Using the same rule we will derive a row for n=4 from the row written for n=3.

Construction of Pascal's triangle

Similarly, from the row 1 4 6 4 1 we can derive the row, which consists of the coefficients of the binomial which are obtained in raising a binomial (a+b) to the power of 5.

Construction of Pascal's triangle

If the row for n=0 (a≠0 or b≠0) is added, then the coefficients of all the rows can be arranged in the form of a triangle.

Construction of Pascal's

In this triangle, the “top sides” consist of ones, and each of the remaining numbers is equal to the sum of the two numbers written above it. This is the triangle known as Pascal’s triangle.

By continuing to record according to the rule noted above, it is possible to obtain the row of coefficients for n=6, 7, etc. in the formula:

Construction of Pascal's

Another pattern seen in Pascal’s triangle:

The sum of the coefficients at n=0, n=1, n=2, etc. is respectively equal to: 20, 21, 22, 23, etc.

In the equality:

Construction of Pascal's

the sum of the coefficients of a polynomial is equal to 2n.

 

In order to raise a binomial to any natural power, there is a formula called Newton’s binomial formula.

Newton's binomial formula

Where,

Then we will obtain:

Construction of Pascal's

The numbers written in each subsequent row of Pascal’s triangle are derived by adding the corresponding numbers of the preceeding row and are the expansion coefficients at the given n.

In doing so, the indexes of power of the number a decrease from n to 0 and the indeces of power of the number b increase from 0 to n.

Examples

 

Example 1

Examples

 

Example 2

Examples