• # Revision:Series

## Introduction

The series of a sequence is the sum of the sequence to a certain number of terms. It is often written as Sn. So if the sequence is 2, 4, 6, 8, 10, ... , the sum to 3 terms = S3 = 2 + 4 + 6 = 12.

## The Sigma Notation

This is best explained using an example:

This is the sum of '3r' for values of 'r' from r = 1 to r = 4.

Another example:

This is the sum of '3r + 2' for r from r = 1 to r = 3.

## The General Case

This is the general case. For the sequence Ur, this means the sum of the terms obtained by substituting in 1, 2, 3, ... n in turn for r in Ur.

In the above example, Ur = 3r + 2 and n = 3.

## Arithmetic Progressions

An arithmetic progression is a sequence which increases by a common difference, d, and which has a first term, a . For example: 3, 5, 7, 9, 11, is an arithmetic progression where a = 3 and d = 2. The nth term of this sequence is 2n + 1 .

In general, the nth term of an arithmetic progression, with first term a and common difference d, is:

.

So for the sequence 3, 5, 7, 9, ...

,

which we already knew.

The sum to n terms of an arithmetic progression

### Example

Sum the first 20 terms of the sequence: 1, 3, 5, 7, 9, ... (i.e. the first 20 odd numbers).

## Geometric Progressions

A geometric progression is a sequence where each term is r times larger than the previous term. r is known as the common ratio of the sequence. The nth term of a geometric progression, where a is the first term and r is the common ratio, is:

For example, in the following geometric progression, the first term is 1, and the common ratio is 2:

1, 2, 4, 8, 16, ...

## The sum of a geometric progression

The sum of a geometric progression is:

### Example

What is the sum of the first 5 terms of the following geometric progression: 2, 4, 8, 16, 32 ?

## The sum to infinity of a geometric progression

In geometric progressions where |r| < 1 , the sum of the sequence as n tends to infinity approaches a value. This value is equal to:

### Example

Find the sum to infinity of the following sequence:

Here, and .

Therefore, the sum to infinity is .

So every time you add another term to the above sequence, the result gets closer and closer to 1.

### Harder Example

The first, second and fifth terms of an arithmetic progression are the first three terms of a geometric progression. The third term of the arithmetic progression is 5. Find the 2 possible values for the fourth term of the geometric progression.

The first term of the arithmetic progression is: a

The second term is: a + d

The fifth term is: a + 4d

So the first three terms of the geometric progression are a, a + d and a + 4d .

In a geometric progression, there is a common ratio. So the ratio of the second term to the first term is equal to the ratio of the third term to the second term. So:

therefore or

The common ratio of the geometric progression, r, is equal to

Therefore, if ,

If ,

So the common ratio of the geometric progression is either 1 or 3 .

We are told that the third term of the arithmetic progression is 5. So:

.

Therefore, when , and when , .

So the first term of the arithmetic progression (which is equal to the first term of the geometric progression) is either 5 or 1.

Therefore, when , and .

In this case, the geometric progression is 5, 5, 5, 5, .... and so the fourth term is 5.

When , and , so the geometric progression is 1, 3, 9, 27, ... and so the fourth term is 27.

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