- Edexcel, Module - C2, Chapter- Sequences and Series
- AQA, Module - C2, Chapter - Sequences and Series
- OCR, Module - C2, Chapter - Sequences and Series

**a) Binomial Expansion 1 b) Binomial Expansion 2,**

**c) Geometric Sequences 1 ,d) Geometric Sequences 2.**

Why we need it ?

Let's say we have to expand the following terms :

(x + y)^1 = x + y

(x + y)^2 = x^2 + 2xy + y^2

(x + y)^3 = (x + y)(x + y)^2

= (x + y)(x^2 + 2xy + y^2)

= x^3 + 3x^2y + 3xy^2 + y^3

As we see, when increasing the power, (n) we get more and more terms, and it becomes more and more confusing, say we had to expand (x + y)^7, this would mean we would get 8 terms, after expanding the bracket 7 times, and then simplifying it. This would be a tedious and extremely long !

The Binomial theorem therefore allows us to expand the

**binomials**( two terms e.g. x + y or x + 3y or a - 3b). In each term of the expansion, there will be a coefficient of the term, these coefficients are either determined by

**Pascal's Triangle**or

**Factorial Formula**.

Let's see how this works, with the examples above... Let's expand (x + y)^3 using the theorem :

Binomial Theorem = To expand the binomial (x + y)^n, it can be written in the form :

This is the binomial coefficient, where n is the power and k is the increasing value till we reach n. |

Let's use take our example using the theorem :

**(x + y)^2**=

**(2,0)**

**x^2 y^0**+

**(2,1)**

**x^1 y^1**+

**(2,2)**

**x^0 y^2**

*The brackets are the binomial coefficients in blue , which are calculated using the formula :

* Notice the x terms start with the power n (in this case 2), and decrease by one power the next term, till the reach the power 0, simultaneously the y terms start with the power 0, and increase by one power the next term, till they reach the power n.

* We finish the binomial expansion when the binomial coefficient is (n,n) in this case (2,2), and when the x term has reached to the power 0 (which is 1), and the y term is to power n (in this case 2) .

__(Remember__)

**anything to the power 0 = 1**and**anything to the power 1 = itself**

Let's simplify it down further, these are the binomial coefficients for each term :

! = Factorial (it calculates the product of all the positive integers less than or equal to some integer n)

(e.g. 4! = 4 x 3 x 2 x 1 = 24)

Note :

**1! = 1**

*and*

**0! = 1**

**(2,0)**= 2! / [ 0! (2 -0)! ]

= 2 / [ 1 x 2 ]

= 2 / 2

= 1

**(2,1)**= 2! / [ 1! (2 -1)! ]

= 2 / [ 1 x 1 ]

= 2 /1

= 2

**(2,2)**= 2! / [ 2! (2 -2)! ]

= 2 / [ 2 x 1]

= 2 / 2

=1

So we have calculated the binomial coefficients for the terms, giving us :

(x + y)^2 = x^2 + 2xy + y^2

__( Second way to find binomial coefficients - helps when n is small)__

**Pascal's Triangle**Another way to find the binomial cofficients, is the Pascal's Triangle. We start with n = 0, and start with number 1. Each row, will list the binomial cofficients in order of the expansion, each starting with 1 and ending with 1, depending on n.

To find the coefficient underneath, two gaps, we simply add the coefficients above it.

**n = 0 1**

**n = 1 1 1**

**n = 2 1 2 1**

**n = 3 1 3 3 1**

**n = 4 1 4 6 4 1**

**n = 5 1 5 10 10 5 1**

* The Triangle goes on as n increases, and the number of coefficients is always ( n + 1)

* The Triangle is symmetrical halfway, so we see that the first half of the coefficients of terms, is always the same as the second half

* Each row of the triangle starts and ends with 1 (meaning the first coefficient and last coefficient of any expansion is always 1)

Double check the expansion (x + y)^2 , where

**n = 2**,

**Check the row n = 2**

The binomial coefficients should be

**1, 2**and

**1**

**1**x^2 +

**2**xy +

**1**y^2

So we've seen how to expand expressions in the form (x + y)^n, and use two methods of working out the binomial coefficients. In the next post I will show you a couple of examples, and another expansion theorem, which allows us to expand algeabraic expressions in the form (1 + x)^ n.

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