Arithmetic (Recursive) Sequences

RECURSIVE

Arithmetic Sequences

Multiple Choice

REVIEW:  What is the explicit equation of this sequence?

• $t\left(n\right)=3n-11$

• $t\left(n\right)=-11n+3$

Explicit Equations

• An explicit formula tells exactly how to find any specific term in a sequence

Recursive Equations

• A recursive formula names the first term (or any other term) and how to get from one term to the next.

Things to know first...

If the term you are on is  $t\left(n\right)$  , then the next term is  $t\left(n+1\right)$  .

For example, if n=1,  $t\left(n\right)$  is  $t\left(1\right)$  , or the first term of the sequence.

Then,  $t\left(n+1\right)$  is  $t\left(1+1\right)$  or  $t\left(2\right)$  , or the second term of the sequence.

So how do we write a recursive equation for this sequence??

Start with a term you are given and the common difference

For example:

 $t\left(4\right)=1$  
This means the fourth term in the sequence has a value of 1.

Then find the common difference.  In this sequence, the common difference is +3.

Now put them together like this!

• The next term in the sequence = the term I know + the common difference

• As an equation...

   $t\left(n+1\right)=t\left(4\right)+3$  

• We know that  $t\left(4\right)=1$  , so we can substitute that into the recursive equation

•  $t\left(4+1\right)\ =\ 1+3$  

•  $t\left(5\right)=\ 4$  

• The fifth term in the sequence has a value of 4!

Let's look at another example before we practice.  Here's my sequence: -2, 1, 4, 7

• Tell me a term that I know by looking at the sequence

• Calculate the common difference

• Put them together in a recursive equation to continue the sequence

•  $t\left(3\right)=4$  ,  $t\left(n+1\right)=t\left(n\right)+3$  

•  $t\left(3+1\right)=4+3$  

•  $t\left(4\right)=7$  

• The fourth term in the sequence has a value of 7!

Multiple Choice

Write a recursive formula for the sequence:

4, 8, 12, 16, 20, ...

• $t\left(n+1\right)=t\left(n\right)+4$

• $t\left(n+1\right)=t\left(n\right)+24$

• $t\left(n+1\right)=t\left(n\right)+2$

• $t\left(n+1\right)=t\left(n\right)-6$

Multiple Choice

Write a recursive formula for the sequence:

-2, 5, 12, 19, 26, ...

•  $t\left(n+1\right)=t\left(n\right)+7$ 

•  $t\left(n+1\right)=t\left(n\right)+19$ 

•  $t\left(n+1\right)=t\left(n\right)+2$ 

• $t\left(n+1\right)=t\left(n\right)-6$

Multiple Choice

Write a recursive formula for the sequence:

27, 15, 3, -9, -21, ...

•  $t\left(n+1\right)=t\left(n\right)-12$ 

•  $t\left(n+1\right)=t\left(n\right)+12$ 

•  $t\left(n+1\right)=t\left(n\right)+4$ 

• $t\left(n+1\right)=t\left(n\right)-6$