Arithmetic Progression Definitions

It is extremely important in this unit that you know the vocabulary associated with arithmetic progressions. The only way to decipher what each question is asking for is by knowing the terminology. With that being said, let's start!

Arithmetic Progression (A.P.)

The logical place to start would be giving the definition of an arithmetic progression! A progression is simply a series of numbers, like so:

1, 6, 3, 198, -4, 15, 0.5, 12.7, 2 ...

The above progression has no observable pattern to it. (Each number seems to be chosen at random.) But look at the next progression below:

3, 7, 11, 15, 19, 23, ...

Do you see a pattern? Can you guess what number comes after 23? If you guessed 27, you're right! The pattern to this progression is that to get each number in the progression, we add 4 to the number that came before it. This is an arithmetic progression, whose definition is given below:

An arithmetic progression (A.P.) is a progression where each number can be found by adding a constant amount to each of the previous terms. (Except for the first term!)

Example

See if you can identify the arithmetic progressions from the progressions below:

A. 5, 7, 9, 11, 13, 15, ...

B. 1, 11, 21, 31, 41, 51, ...

C. 2, 4, 8, 16, 32, 64, ...

D. 1, 1.5, 2, 2.5, 3, 3.5, ....

E. 9, 6, 3, 0, -3, -6, -9, -12, ...

Solution

A. A IS an arithmetic progression. Notice that to get to each sucessive number you add a constant amount (in this case you add 2.)

B. This IS an arithmetic progression. Again, we have to add 10 (the constant amount) to each number to get the next number.

C. This is NOT an arithmetic progression. You cannot get from number to number just by adding a constant value. (First you add 2, then add 4, then add 8, etc...). This is an example of a geometric progression which you will study in Pre-Calculus 40S.

D. This IS an arithmetic progression. The constant value being added in this case is 0.5.

E. This is a tricky one! This actually IS an arithmetic progression. The constant value being 'added' is -3! (Watch: 9 + (-3) = 6...6 + (-3) = 3, etc.) Not all arithmetic progressions have to increase in value.

Common Difference

The common difference in an arithmetic progression is the constant value that you add to each number to get to the next number. For example, in the following A.P.:

4, 9, 14, 19, 24, 29, ...

the common difference is 5. You can ALWAYS find the value of the common difference by taking any number in the progression and subtracting the number before it. To illustrate this, look again at the A.P. above. To calculate the common difference, do ANY of the following:

24 - 19 = 5
or
19 - 14 = 5
or
9 - 4 = 5.

Again, to find the common difference:

Take any number in the A.P. and subtract the number immediately before it.

Example

Find the common difference in the following A.P.s:

A. 3, 15, 27, 39, ...

B. 15, 11, 7, 3, -1, -5, ...

C. , ...

Solution

A. The common difference here is 12. You can find this by (for example) the calculation 27 - 15.

B. The common difference is -4. This can be found by (for example) 3 - 7.

C. This one is a little trickier, especially if you are uncomfortable with fractions. To find the common difference, take any number and subtract the number immediately before it. For example, we could do: to get a common difference of

Term (Including n, the index number)

The last definition we are going to look at is the definition of term:

A term is one of the numbers in an arithmetic progression.

For example, this A.P. has 5 terms listed: 6, 9, 12, 15, 18, ... . The number of terms in this A.P. is infinite, but there are 5 terms shown.

A term is usually denoted as , where the n gets replaced with a whole number. The n is called the index number and is used to denote which term you are currently examining. For example, refers to the FIRST term of a progression, the SECOND term, the FIFTH term, and so on.

Example

1. In the arithmetic progression 5, 8, 11, 14, 17, ... give the value of .

2. In the arithmetic progression -6, 0, 6, 12, 18, 24, ... how many terms are shown? What is the value of ?

3. In the arithmetic progression 1, 3, 5, 7, 9, ... what is the value of ?

Solution

1. The value of is 17.

2. There are 6 terms shown, and the value of is -6.

3. The value of is not shown, but we can figure it out! The common difference is 2, so to find the value of we just have to add 2 to (which is 9). So logically, must be 11.

What if in the 3rd example above they had asked for ? Would you have to add 2 again and again and again until you reached the 115th term? Although this is A method for finding , there are better and quicker ways to find that value. These alternative methods are what you will study in the rest of the unit. Now you can either move on to the next topic (recursion formulas) or head back to the Arithmetic Progressions main page.