**Binomial Expansion**

(a+b)^{n} is called a **binomial** because it has two terms.

Consider the expansions (a+b)^{1}, (a+b)^{2}, (a+b)^{3}, (a+b)^{4} (yes, let’s fully expand them and inspect the result). If you look at the coefficients of these do you notice any pattern?

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**Binomial coefficients**

The numbers in Pascal’s triangle are actually the binomial coefficients , where n is the row of Pascal’s triangle and k goes from 1 to n from left to right in the triangle.

We can calculate these manually, or using the calculator’s buttons ‘‘ or ‘!’

In general, for all **natural numbers** n:

It is worth noticing that for all n, , , and for all n and k:

**Worked Examples**

**Exercise**

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**Arithmetic progression**

- We use the following notation:
**a**or**a**refers to the_{1}**first term**;**d**refers to the**common difference**;**l**referes to the**last term**.

Note that the common difference may be zero or a negative number.

The nth term is a + (n-1)d.

We use a method developed by Gauss to derive the formula for the sum of the first n terms in an arithmetic sequence. (Let’s do it!)

We should also be aware that the nth term in an arithmetic sequence is S_{n}-S_{n-1}

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**Exercise**

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**Geometric Progression**

- We use the following notation:
**a**or**a**refers to the_{1}**first term**;**r**refers to the**common ratio**;

The nth term is ar^{n-1}

We can multiply the whole sum by r and then subtract this to find the formula for the sum of n terms (Let’s do this!)

**Worked Examples**

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**Infinite Geometric Progressions**

If the **absolute value** of the **common ratio** is less than 1, then an infinite geometric series will **converge**.

S_{∞} =

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**Combining Arithmetic Series and Geometric Series in a Question**

**Worked Example**

Below is an extract from the formula book “MF19” that shows all the relevant formulae to this topic that you will be given in the exam:

**Exercise & General Exercises**

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