# Series

Source: https://projectalevel.co.uk/maths/algebra/series

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#### **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:

4
S 3r
r = 1

This is equal to 3×1 + 3×2 + 3×3 + 3×4 = 3 + 6 + 9 + 12 = 30.

3
S 3r + 2
r = 1

This is equal to:
(3×1 + 2) + (3×2 + 2) + (3×3 + 2) = 24 .

The General Case:

n
S Ur
r = 1

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: a + (n - 1)d . So for the sequence 3, 5, 7, 9, ... Un = 3 + 2(n - 1) = 2n + 1, which we already knew.

#### **The sum to n terms of an arithmetic progression**

Sn = ½ n \[ 2a + (n - 1)d \]

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

S20 = ½ (20) \[ 2 × 1 + (20 - 1)×2 \]
= 10\[ 2 + 19 × 2\]
= 10\[ 40 \]
= 400

#### **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:

arn-1

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:

a(1 - rn)
 1 - r

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

S5 = 2( 1 - 25)
 1 - 2

= 2( 1 - 32)
 -1

= 62

**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:

a
1 - r

Example:
Find the sum to infinity of the following sequence:

1 , 1 , 1 , 1 , 1 , 1 , ...
2 4 8 16 32 64

Here, a = 1/2 and r = 1/2
Therefore, the sum to infinity is 0.5/0.5 = 1 .
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:

a + d = a + 4d
 a a + d

(a + d)(a + d) = a(a + 4d)
a² + 2ad + d² = a² + 4ad
d² - 2ad = 0
d(d - 2a) = 0
therefore d = 0 or d = 2a

The common ratio of the geometric progression, r, is equal to (a + d)/a
Therefore, if d = 0, r = 1
If d = 2a, r = 3a/a = 3
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 a + 2d = 5 . Therefore, when d = 0, a = 5 and when d = 2a, a = 1 .
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 d = 0, a = 5 and r = 1. In this case, the geometric progression is 5, 5, 5, 5, .... and so the fourth term is 5.When d = 2a, r = 3 and a = 1, so the geometric progression is 1, 3, 9, 27, ... and so the fourth term is 27.