Arithmetic (Recursive) Sequences

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

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

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

As an equation...

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!

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!