- Define arithmetic sequences and series.
- Use of the formulae for the nth term and the sum of the first n terms of the sequence.
- Use of sigma notation for sums of arithmetic sequences.

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##### 0. Sequences in general - Series.

**SEQUENCE**

### A sequence is just an ordered list of numbers (terms in a definite order). For example :

### Usually, the terms of a sequence follow a specific pattern, for example

#### \(0,2,4,6,8,10,…\) (even numbers, \(u_n=2n-2; n \in\mathbb{N}^{*} \))

#### \(1,3,5,7,9,11,…\) (odd numbers, \(u_n=2n-1; n \in\mathbb{N}^{*} \))

#### \(5,10,15,20,25,…\) (positive multiples of 5, \(u_n=5n; n \in\mathbb{N}^{*} \))

#### \(2,4,8,16,32,…\) (powers of 2, \(u_n=2^n; n \in\mathbb{N}^{*} \))

### We use the notation un to describe the \(n^{th}\) term. Thus, the terms of the sequence are denoted by

#### \( u_{1} ; u_{2} ; u_{3} ; u_{4} ; u_{5} ; …\)

**SERIES**

### A series is just a sum of terms:

### \(S_{\color{blue}{n}}=u_{1}+u_{2}+u_{3}+…+u_{\color{blue}{n}}\) : The sum of the firts n terms

### \(S_{\color{green}{∞}}=u_{1}+u_{2}+u_{3}+\color{green}{…}\) : The sum of all terms

### We say that \(S_{∞}\) is an infinite series, while the finite sums \(S_{1}\), \(S_{2}\), \(S_{3}\),…are called partial sums.

### Example :

### Consider the sequence of odd numbers :

### \(1,3,5,7,9,11,…\)

### Some of the terms are the following

### \(u_1 =1\), \( u_2=3\), \( u_3 =5\), \(u_6 =11\), \(u_{10} =19\)

### Also,

### \(S_1 =1\),

### \(S_2 =1+3=4\),

### \(S_3=1+3+5=9\),

### \(S_4 =1+3+5+7=16\)

### Finally,

### \(S_∞ =1+3+5+7 +…\)

**SIGMA NOTATION \(\sum\limits_{n=1}^{k}\)**

### Instead of writing :

### \(u_{\color{red}{1}}+ u_2 + u_3 + u_4 + u_5 +u_6 +u_{\color{green}{7}} \)

### We may write :

### \(\sum\limits_{n=\color{red}{1}}^{\color{green}{7}}u_n\)

### It stands for the sum of all terms \(u_n\) , where \(n\) ranges from 1 to 7.

### In general,

### \(\sum\limits_{n=1}^{k}u_n\)

### expresses the sum of all terms \(u_n\) , where \(n\) ranges from 1 to k.

### We may also start with another value for \(n\), instead of 1, e.g. \(\sum\limits_{n=4}^{11}u_n\)

### Examples :

### \(\sum\limits_{n=1}^{3}2^n=2^1+2^2+2^3=2+4+8=14\)

### \(\sum\limits_{k=3}^{6}(2k+1)=7+9+11+13=40\)

**NOTICE**

### There are two basic ways to describe a sequence

### A) by a **GENERAL FORMULA**

### We just describe the general term \(u_n\) in terms of \(n\).

### For example, \(u_n= 2n\)

### It is the sequence \( 2,4,6,8,10,… \)

### (It gives \(u_1 = 2; u_2= 4; u_3 = 6; … \))

### B) by a **RECURSIVE RELATION** *(mainly for Math HL)*

### Given: \(u_1\) ,

### the first term \(u_{n+1}\) in terms of \( u_n\)

### For example,

### \(u_1 = 7\)

### \(u_{n+1} = u_n + 2\)

### This says that the first term is 7 and then

### \(u_2 = u_1 +2\)

### \(u_3 = u_2 +2\)

### \(u_4 = u_3 +2 \) and so on.

### In simple words, begin with 10 and keep adding 2 in order to find the following term.

### It is the sequence \(7, 9, 11, 13, 15, …\)

##### 1. Arithmetic sequences

## Definition:

**Arithmetic sequences** are characterized by the fact that to get from one term to the next we always add the same amount.

### The amount we add is known as the common difference and is usually referred to as *d*, \( d \in \mathbb{R}\)

For example :

The sequence whose first few terms are:

is arithmetic, with common difference \( d=4 \)

### So, we need two numbers to find the others terms

### The first term **\( u_1 \)**

### The common difference **\(d\)**

Examples :

If \( u_1 =1, d=2\) the sequence is \(1 ; 3 ; 5 ; 7 ; 9 ; …\)

If \( u_1 =-10, d=5\) the sequence is \(-10 ; -5 ; 0 ; 5 ; 10 ; …\)

If \( u_1 =10, d=-3\) the sequence is \(10 ; 7 ; 4 ; 1 ; -2 ; …\)

**What is the general formula for \(u_n\) ?**

Let us think:

In order to find \( u_3\) , we start from \(u_1\) and then add **2** times the difference \(d\). \(u_{\color{orange}{3}}=u_1+\color{orange}{2}d\)

Hence,

\(u_{\color{orange}{4}} = u_1+\color{orange}{ 3}d\)

\(u_{\color{orange}{5}} = u_1+ \color{orange}{4}d\)

Similarly,

\(u_{\color{orange}{10}} = u_1+ \color{orange}{9}d\)

\(u_{\color{orange}{50}} = u_1+ \color{orange}{49}d\)

In general,

\(u_n = u_1+ (n-1)d\)

### The nth term of an arithmetic sequence is given by:

### \(a_n=a_1+(n−1)d\)

### where \(a_1\) is the first term and \(d\) the common difference.

##### 2. Sum of an arithmetic sequence

**Gauss Problem**

In elementary school in the late 1700’s, Gauss was asked to find the sum of the numbers from 1 to 100. The question was assigned as “busy work” by the teacher, but Gauss found the answer rather quickly by discovering a pattern. \(1+2+3+4+5+6+ ….+97+98+99+100 = ? \) |

The answer is 5050. How can you do it without a calculator in a matter of minutes like Gauss?

### The sum \(S_n\) of the first \(n\) terms of an arithmetic sequence is given by :

### \(S_n=(u_1+u_n)\times \frac{n}{2}\)

### \(S_n= (2u_1+ (n-1)d)\times\frac{n}{2}\)

### \( u_1\) is the first term,

### \(u_n\) the last term,

### \(d\), the common difference.

##### 3. Exercises

##### 4. MCQ

##### Teachers Ressources

Quizizz as a warm up or at the end of your course. This resource can be used by the students themselves by clicking on the “practice” button:

– Arithmetic sequence : definitiion and common difference – Arithmetic sequence : formula – test |