# A-level Mathematics/OCR/C2/Sequences and Series

## Definitions

[edit | edit source]A **sequence** is simply a list of numbers in a particular order. We call these numbers the **terms** of the sequence. For instance, 2,4,6,8 are the first four terms in the sequence of even positive integers. When we take the sum of the terms in a sequence, we get a **series**. For example, 2+4+6+8+... is a series.

We denote the terms in a sequence by where is the number of the term in question. For example, we have , , , and so on, in the sequence described above.

A **definition** is a rule that tells us how to compute each term in a sequence. For example, a rule for the sequence above is . A **relation** describes how each term is related to other terms. For instance, a relation for the above sequence is .

## Sigma () Notation

[edit | edit source]As you might have suspected, describing a series with the help of some of its terms isn't always a good idea --- if too few terms are used, the series can be ambiguous to your reader; on the other hand, you risk insulting your reader by writing out too many terms! To express a series succinctly, we use the **sigma notation** instead.

In general, a series may be written as , which means "sum of all terms beginning with up to and including ". Hence,

.

As an example, the series 2+4+6+8+... may be written as .

## Recognising Simple Progressions

[edit | edit source]A **progression** is just another word for a **sequence**. In this module, you are expected to be well-acquainted with two very common types of progressions --- the **arithmetic progression** and the **geometric progression**.

Briefly, an arithmetic progression or **AP** is a sequence in which each successive term is the *sum* of the previous term and a fixed value. An example of an AP is 1,4,7,10,..., where the difference between successive terms is 3.

A geometric progression or **GP** is a sequence in which each successive term is the *product* of the previous term and a fixed value. An example of a GP is 2,4,8,16,32,..., where each term is twice the value of the previous term.

## Arithmetic Progression (AP)

[edit | edit source]An arithmetic progression (AP) is a sequence that can be written in the following way: , where are constants. The first term in the AP is denoted by , and the **common difference** between subsequent terms is denoted by . Thus, the series 1,4,7,10,..., is an arithmetic progression with and .

### Rules

[edit | edit source]The common difference can be calculated by , where .

The th term is given by .

The sum of the first terms of an AP (with as its first term and as its last term) is given by .

In fact, more generally, the sum of consecutive terms in an AP is given by .

### Example

[edit | edit source]*What is the sum of the even numbers 2, 4, 6, 8, ..., 100?*

The given sequence can be expressed as an AP with , and . We want the sum of the first 50 terms of the AP:

.

## Geometric Progression (GP)

[edit | edit source]A geometric progression (GP) is a sequence that can be written in the following way: , where are constants. The first term in the GP is denoted by , and the **common ratio** between subsequent terms is denoted by .

### Rules

[edit | edit source]The common ratio can be calculated by , where .

The th term is given by .

The sum of the first terms of an GP is given by .

Proof of this is given by:

### Sum Of An Infinite Geometric Series

[edit | edit source]We say that the geometric series is **convergent** if the **sum to infinity** approaches some limit. This occurs when . Hence, if , then

.

Proof of this is given by:

## Binomial expressions

[edit | edit source]A binomial is a polynomial with two parts in the form , such as . When a binomial is raised to a power, you could simplify it by multiplying out the brackets several times. The expanded polynomial is called a **binomial expansion**, and all binomial expansions follow a pattern that can be used to expand binomials quicker than multiplying out several brackets. For now, we will only look at binomial expressions which are raised to positive integers.

### Expansions of

[edit | edit source]Here are the expansions of raised to different powers.

If you look at the coefficient of each term, you may notice a pattern. These numbers are called **binomial coefficients** and are found by adding the two numbers above it.

### Pascal's triangle

[edit | edit source]Binomial coefficients are more commonly known as **Pascal's triangle**, named after Blaise Pascal.

The first 10 lines of Pascal's triangle are:

(1) 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 1 7 21 35 35 21 7 1 1 8 28 56 70 56 28 8 1 1 9 36 84 126 126 84 36 9 1 1 10 45 120 210 252 210 120 45 10 1 1 11 55 165 330 462 462 330 165 55 11 1

Since each number is found by adding the two numbers above it, it is possible to find a few lines of the triangle to help you expand binomials. For binomials raised to powers greater than 10, you should use the binomial coefficient formula.

### Binomial coefficient formula

[edit | edit source]When a binomial is raised to a large power, it may be too time consuming to find the binomial coefficients by writing out Pascal's triangle. Fortunately, there is a formula that can find any line of Pascal's triangle.

If is the power of the expansion, and is the number of the term in a single row, the binomial coefficient formula is:

The means **factorial** and multiplies by every integer less than itself, down to 1.

So .

To find the binomial coefficients, you use the formula with the required value of , and , , , and so on, until .

Most scientific calculators will have two buttons that will be useful in this process, one is the factorial button, usually labelled ** n!** and the other will actually find and is often labelled

**or**

*n*C*r***. (The C stands for "choose" or "combination" which is based on the formula's use in probability.)**

You should be aware that Pascal's triangle is symmetrical, so once a coefficient is repeated, you can write down the rest of the coefficients with ease.

### Expanding binomials

[edit | edit source]Now that you know how to find the coefficients in a binomial expansion, you can easily expand any binomial that is raised to a positve integer by following these simple steps:

For a binomial in the form ,

- Write down in descending powers, from to
- Write down in ascending powers, from to , making sure that you place the terms so that the powers add up to
- Add the binomial coefficients to each term, either from line in Pascal's triangle (ignoring the 1 at the top), or by using the binomial coefficient formula.

You then simplify where necessary.

For example, for the expansion of :

in descending powers:

in ascending powers:

Grouping everything together we now have:

Adding in the binomial coefficients:

Finally simplifying will now give us:

$16 $

This process is summarised in the equation known as the binomial theorem:

In case you are not familiar with sigma notion this means:

Several simplifications can be made but they aren't worth memorising as you will pick them up automatically:

*This is part of the C2 (Core Mathematics 2) module of the A-level Mathematics text.*

Dividing and Factoring Polynomials / Sequences and Series / Logarithms and Exponentials / Circles and Angles / Integration