SOLUTION: My homework is asking me to find a recursive definition for a sequence, but we didn't do anything like this in class so I'm at a loss. If you can, please explain how one would find

First we observe the pattern

a(1)=1
a(2)=7
a(3)=13
a(4)=19
a(5)=25
a(6)=31
a(7)=37
a(8)=43

a(1) =  1                    <--the first term
a(2) =  7 =  1+6 = a(1) + 6  <--to get the 2nd term, add 6 to the 1st term. 
a(3) = 13 =  7+6 = a(2) + 6  <--to get the 3rd term, add 6 to the 2nd term.
a(4) = 19 = 13+6 = a(3) + 6  <--to get the 4th term, add 6 to the 3rd term.
a(5) = 25 = 19+6 = a(4) + 6  <--to get the 5th term, add 6 to the 4th term.
a(6) = 31 = 25+6 = a(5) + 6  <--to get the 6th term, add 6 to the 5th term.
a(7) = 27 = 31+6 = a(6) + 6  <--to get the 7th term, add 6 to the 6th term.
a(8) = 43 = 37+6 = a(7) + 6  <--to get the 8th term, add 6 to the 7th term.

So if we have

a(2) = a(1) + 6
a(3) = a(2) + 6
a(4) = a(3) + 6
a(5) = a(4) + 6
a(6) = a(5) + 6
a(7) = a(6) + 6
a(8) = a(7) + 6

We notice that the red numbers go 1,2,3,4,5,6,7,8.
So they could be represented by the letter n

We notice that the blue numbers go 2,3,4,5,6,7,8,9.
Since they are always 1 more than the red numbers,
they could be represented by n+1

So each equation above could be represented by

a(n+1) = a(n) + 6

A recursion formula consists of a part to get the sequence
started and an equation to tell how to get the next term
from what went before.  So the recursion formula is:

a(1) = 1,  a(n+1) = a(n) + 6    <--the recurstion formula

This recursion formula tells us:

The first term is 1, and the (n+1)th term is the nth term + 6.

Edwin