# At a Glance - Arithmetic Sequences

An arithmetic sequence is a sequence where the step from one term to the next is constant. That is, you always add the same thing to get from one term to the next.

An arithmetic sequence is like going up a huge flight of stairs. Most days, you'll walk up the flight of stairs one step at time. Sometimes, though, you are in a hurry, and you skip stairs as you rush up the stairs to get your coat before the bus comes. The increment between each of your steps is constant: one stair if you are on time, and two stairs if you are in a hurry. Be careful not to trip. A bruised shin is the worst.

### Sample Problem

The sequence

2, 4, 6, 8, 10, ...

is an arithmetic sequence. To get from one term to the next, we always add 2.

### Sample Problem

The sequence

2, 4, 8, 16, ...

is not an arithmetic sequence. To get from 2 to 4 we add 2, but to get from 4 to 8 we add 4.

Since we didn't add the same thing every time, this isn't arithmetic.

Saying the step up from one term to the next term is constant is the same as saying the step down from one term to the previous term is constant. In an arithmetic sequence, the difference between one term and the previous term must always be the same. Our base-running analogy breaks down here. You wouldn't want to run them backwards unless you go from first to second base facing the wrong way.

### Sample Problem

Is the sequence

4, 8, 12, 16, ...

arithmetic?

Let's find the difference between each pair of successive terms.

8 – 4 = 4

12 – 8 = 4

16 – 12 = 4

The difference between each pair of consecutive terms is 4.

That means this is an arithmetic sequence.

### Sample Problem

Is the sequence

3, 6, 10, 15, 21, ...

arithmetic?

Let's find the difference between each pair of terms.

6 – 3 = 3

10 – 6 = 4

We don't have to go any further. The difference between successive terms isn't always the same. That means this is not an arithmetic sequence.

It's also okay for the step from one term to the next to be negative, as long as the step is constant.

### Sample Problem

The sequence

10, 0, -10, -20, ...

is an arithmetic sequence because the step from one term to the next is always -10.

Now that we're clear on what an arithmetic sequence is, let's put the definition into symbols and an equation. Mathematicians love equations.

In an arithmetic sequence, to get from one term to the next term we add some constant d.

In symbols,

an + 1 = an + d.

This is the same as saying that to get from one term to the previous term we subtract some constant d.

In symbols,

an + 1an = d.

Arithmetic sequences are usually defined in terms of subtraction rather than addition. The value d is called the common difference for the sequence.

We'll start all our arithmetic sequences with n = 1 corresponding to the first term, just because we can. We can also jump up on our desks and sing, "Take Me Out to the Ball Game," at the top of our lungs, but it probably won't help us with sequences.

An arithmetic sequence is completely determined by two things: its starting term a1 and its common difference d. Once you know where an arithmetic sequence starts and what its step size is, you know everything there is to know about it.

If we throw a baseball backwards behind us, we might break our mother's chandelier. But if we know two terms in an arithmetic sequence, we can work backwards to figure out a1 and d.

We can figure out a1 and d even if we're given two terms am and an that aren't consecutive (assume m < n). We have to do one extra operation. After finding the difference an am, we have to divide by the number of steps required to get from am to an. That gives us d, and we can proceed as before.

This means we need to know how many steps it takes to get from one term am to another term an.

#### Example 1

 Write the first four terms of the arithmetic sequence with a1 = 4 and d = 5.

#### Example 2

 If a1 = 3 and d = 5, find a5.

#### Example 3

 If a1 = 12 and d = -2, find a11.

#### Example 4

 An arithmetic sequence has a5 = 11 and a6 = 13. Find a1 and d.

#### Example 5

 An arithmetic sequence has a10 = 52 and a11 = 56. Find a1 and d.

#### Example 6

 Find a1 and d for the arithmetic sequence with a2 = 8 and a4 = 28

#### Example 7

 Find a1 and d for the arithmetic sequence with a50 = 14 and a100 = -136.

#### Exercise 1

Determine whether the sequence is arithmetic or not.

1, 2, 3, 4, 5,...

#### Exercise 2

Determine whether the sequence is arithmetic or not.

10, 13, 16, 20,...

#### Exercise 3

Determine whether the sequence is arithmetic or not.

1, 3, 5, 7, 9,...

#### Exercise 4

Determine whether the sequence is arithmetic or not.

15, 21, 26, 30,...

#### Exercise 5

Determine whether the sequence is arithmetic or not.

0, 10, 20, 30,...

#### Exercise 6

Determine whether the sequence is arithmetic or not.

1, -1, 1, -1,...

#### Exercise 7

Determine whether the sequence is arithmetic or not.

5, -10, 15, -20,...

#### Exercise 8

Determine whether the sequence is arithmetic or not.

0, -2, -4, -6,...

#### Exercise 9

Write the first four terms of the arithmetic sequence with

a1 = 10 and d = 7

#### Exercise 10

Write the first four terms of the arithmetic sequence with

a1 = 5 and d = -2

#### Exercise 11

Write the first four terms of the arithmetic sequence with

a1 = -1 and d = -3

#### Exercise 12

How many steps does it take to get from

a1 to a6?

#### Exercise 13

How many steps does it take to get from

a1 to a75?

#### Exercise 14

How many steps does it take to get from

a1 to an?

#### Exercise 15

Find the requested term for the arithmetic sequence.

Find a8 if a1 = 20 and d = 3.

#### Exercise 16

Find the requested term for the arithmetic sequence.

Find a11 if a1 = 1 and d = -1.

#### Exercise 17

Find the requested term for the arithmetic sequence.

Find a20 if a1 = 5 and d = 2.

#### Exercise 18

Find the requested term for the arithmetic sequence.

Find a10 if a1 = -3 and d = -5.

#### Exercise 19

Find the requested term for the arithmetic sequence.

Find a100 if a1 = 7 and d = 5.

#### Exercise 20

Find a1 and d for the arithmetic sequence with the given terms.

a100 = 204, a101 = 206

#### Exercise 21

Find a1 and d for the arithmetic sequence with the given terms.

a11 = 22, a12 = 33

#### Exercise 22

Find a1 and d for the arithmetic sequence with the given terms.

a20 = -2, a21 = -7.

#### Exercise 23

How many steps does it take to get from a10 to a15?

#### Exercise 24

How many steps does it take to get from a100 to a200?

#### Exercise 25

How many steps does it take to get from am to an (assume m < n)?

#### Exercise 26

Find a1 and d for the arithmetic sequence with the given terms.

a20 = 13, and a25 = 33