Recursion Formulas

As was mentioned in the lesson on definitions, we are going to start developing formulas to help us work with arithmetic progressions. The first type of formula that is examined is a recursion formula.

A recursion formula is just a mathematical way of describing how each term in an A.P. is connected to the previous term. For example, look at the following A.P.:

2, 7, 12, 17, 22, ...

In this A.P. the common difference is 5. This means in english that:

To find ANY term of this A.P., add 5 to the value of the previous term.

The above sentence can be expressed mathematically as:

The formula may look confusing at first. Let's examine the parts more carefully. means "term number n". Which term is term n? Any term you want. is just another way of saying "ANY term of the progression". You will fill in the value for n depending on which term you are looking for.

• means "term number n". Which term is term n? Any term you want. is just another way of saying "ANY term of the progression". You will fill in the value for n depending on which term you are looking for.
• means "the term immediately before term n". How do we get this? Well, in the formula above, if you were looking for , the value of n is set to 6. (You are looking for the value of the sixth term.) So evaluates to (remember that n = 6 for now) or . is the term that immediately precedes (comes before) term 6 ().

So let's look at that formula again:

(ANY term) = (the term before it) + 5

To find any term of this A.P., add 5 to the value of the previous term.

Let's try it out! Using the recursion formula for this A.P. (2, 7, 12, 17, 22, ...), find the value of .

Fill in the value of n into the formula (we are looking for the 6th term so n = 6):

simplify the subscript on the right hand side term:

We know the value of . is equal to 22. So substitute that value into the formula and solve:

= 22 + 5
= 27

In general, a recursion formula looks like this:

where d is the common difference for the specific A.P. that you are working on.

The other thing you will have to be able to do with recursion formulas is to write an A.P. when you are given it's recursion formula. The trick to these questions is that you MUST be given the value of the first term since there is no term BEFORE the first term to help you find the value of the first term. Look at the example below:

Example

Find the first five terms of the A.P. that has a first term () of 7, and the following recursion formula: .

Solution

You know the first term of the A.P. - it was given in the question. So you know that this A.P. starts with the number 7 as the first term:

7, ...

Now you have to decipher the recursion formula. The recursion formula says (in English) "To find ANY term of the sequence, find the term before it and add 12." So, to find the second term we take term 1 and add 12 to it. If you use the formula it would look like this:

and simplifying...

so therefore...

= 7 + 12
= 19

So the second term of this A.P. is 19. Now we have:

7, 19, ...

Do you HAVE to use the formula to find each successive term? NO! Once you have deciphered what the formula means (basically, add 12) you can just do what it says and add 12 to each term to get the next term. So the first five terms would be:

7, 19, 31, 43, 55

Recursion formulas don't always have to generate arithmetic progressions. They can generate all types of weird progressions. Look at the following example:

Example

Find the first five terms of the progression defined by the recursion formula , with = 3.

Solution

Again, it is easy to find the value of the first term, since you are told that it is 3. So this progression starts like:

3, ...

Now figure out what the formula says in English. Basically, it says that any term () is equal to the previous term () squared. So, IF you use the formula to find the second term, it would look like this:

and...

Finally, filling in the values gives us...

= 3²
or...
= 9

So now we have...

3, 9, ...

Can you figure out what the next term should be without using the formula? The third term is equal to the second term squared. So the thrid term is 9² which is 81. This progression gets large quickly! The final solution looks like this:

3, 9, 81, 6561, 43046721, ...

Now that you have seen some examples involving recursion formulas, you can try some sample questions or head back to the Arithmetic Progressions main page.