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Objectives: :

• 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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## Definition:

### 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 $$d=4$$

### 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,

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

### 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?

### $$d$$, the common difference.

##### 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:

 – test