*Sequence* * is called the arithmetic sequence if every member of that sequence is equal to the previous member increased by a constant d in such way that:*

Number is called the ** difference of arithmetic sequence**.

Arithmetic sequence is uniquely identified by its first member and the difference .

Why does arithmetic sequence have that name? Every member of arithmetic sequence, except for the first is arithmetic middle of two adjacent members.

**Example 5**. Write first five members of arithmetic sequence whose first member is and whose difference is .

Here you can also test previous statement:

Arithmetic sequence has few divisions, depending on the difference.

**If then the sequence is rising.**

For

We say that the sequence is rising if the members are getting larger and larger.

**If then the sequence is falling.**

For

We can say that the sequence is falling if the members are getting smaller and smaller.

**If then the sequence is a constant sequence.**

For

## General member of arithmetic sequence

To find general member of arithmetic sequence we have to observe the definition of a sequence and try to extract every member through the elements that have to be given in the task. This means that we’ll try to represent every member as a relationship between the first member and the difference.

…

Using the same logic we got the nth member.

*The general member of arithmetic sequence whose first member is ** with difference has a form:*

**Example 6**. Find tenth member of a sequence

From the task itself we can see the first member , and calculate difference .

Now for the tenth member we simply insert everything in the form of a general member we know.

**Example 7.** If fifth member of an arithmetic sequence is , and the tenth is , what is the first member of the sequence and what is the difference of this sequence?

This now comes down to a simple system of equations. By the method of contrary coefficients:

**Sum of first n members of arithmetic sequence:**

If we have a given arithmetic sequence . If we mark the sum of first members with then:

….

How can we determine without having to add together all members one by one?

Let’s take it step by step.

If we add together these two equalities we get:

*The sum of first members **is given with a formula** *

As we know, so we can write this formula as:

**Example 8**. Determine the sum of the first fourteen members of the sequence .

### For Those Who Want To Learn More:

- Mathematical reasoning
- Fractions
- Vectors
- Naming decimal places
- Form of quadratic equations, discriminant formula, Vieta’s
- Graphs of trigonometric functions
- Geometric sequence
- Methods of solving trigonometric equations and inequalities
- Definition, properties and graphing of absolute value
- Basic trigonometric functions