## Arithmetic Sequences

In **mathematics**, a **sequence** is an orderly set of numbers. Each element in a sequence is called **term**.

10, 15, 20, 25, …

10, 20, 40, 80, …

The times tables show some simplest sequences:

1,2,3,4,5, …

2,4,6,8,10, …

3,6,9,12,15, …

4, 8, 12, 16, 20 …

These three dots “…” are called ellipsis, and means that the sequence goes on forever (**infinite sequence**).

For example, **4, 8, 12, 16, 20 … **make up the sequence that begins with the number 4, and increases by 4 each time. 4 is the first term, 8 is the second term, 12 is the third term, and so on.

An ARITHMETIC sequence has a common difference between terms. In this example, the** common difference** is **4**.

After finding out the difference between the numbers, it becomes quite easy to identify additional terms in the sequence. Adding 4 to the previous number helps to arrive at the next number in the sequence, and thereby creates a pattern.

## Terms in an arithmetic sequence:

**a, a+d, a+2d, a+3d, … , a+(n-1)d**

The first term is **a**, the common difference is **d**, and the number of terms is **n**.

Real world example:

The arithmetic sequence 4, 8, 12, 16, 24 … represents the total number of milk quarts that a store sold, after each additional gallon of milk is bought by the consumers.

**RULES FOR SEQUENCES AND PATTERNS**:

**Find the Missing Numbers**__. __

How may we find the missing numbers in a sequence?

First, we must identify a rule that relates to the sequence. From time to time, if we just look at the numbers we can see a pattern, thus the number sequence.

**Pattern. **A design (geometric) or sequence (numeric or algebraic) that is predictable because some aspect of it repeats ■□□□■□□□■□□□■□□□

Example 1: Toothpicks Pattern

The following pattern may be made with toothpicks, one for each straight line.

1 triangle = 3 toothpicks, 2 triangles = 5 toothpicks, 3 triangles = 7 toothpicks, ….

Each time we add one triangle, we need to add two more toothpicks. To add another triangle, we need to add two more toothpicks

**Sequence 1: ** 3, 5, 7, 9, …

**Difference:** 2

**Pattern:** “add 2 to the previous number to get the next number”

**Question:** *What are the next two terms in the sequence?*

**Answer:** *The next two terms are 9+2 = 11 and 11+2 = 13*

**Question**: Write an expression that can be used to find the **n ^{th}** term of the sequence 3, 5, 7, 9, …

**Answer**: The first term is **a=3**, the common difference is **d=2**, the **n ^{th}** term of the sequence is

**a+(n-1)d**= 3+(n-1)2 = 3 + 2n -2 =

**2n+1**

**Question**: Find the 10^{th} term of the sequence 3, 5, 7, 9, …

**Answer**: The first term is **a=3**, the common difference is **d=2**, the 10th term of the sequence is **a+(n-1)d **= 3+(10-1)2 = 3+(9)2 =** 21**

**Enrichment:** Find the expression for the n^{th} term, in the following pattern of squares made with toothpicks.

## Graphing *Arithmetic Sequences*

When we graph the points of an arithmetic sequence, they lie on a straight line.

All arithmetic sequences have linear graphs.

**Example**: A fair charges an entry fee of $6 and each ride is $2. The expression **6+2n** tells us how much money we need to play **n** games.

The *graph* below shows the *arithmetic sequence 6+2n*

** **

**Arithmetic Sequence as a Linear Function**

There is much resemblance between arithmetic sequences and linear functions.

An arithmetic sequence is a linear function whose domain is the set **natural numbers** (positive integers: 1, 2, 3, …), and the common difference is equivalent to the slope. However, an arithmetic sequence does not have a y-intercept (because **n** cannot be zero).

Linear functions are lines that continue forever in each direction, hence its domain is all real numbers (including integers, rational and irrational numbers).

Linear Relationships (Y=5X+20)

Linear Relationships (Y=2X+10)

Linear Relationships (Y=500X+2000)

Linear Relationships (Y=100X+200)

Linear Relationships (Y=100-5X)

Linear Relationships (Y=0.2X + 10)

Linear Relationships (Y=6 – X)