The starting salary at a company is $42,000 per year. The company automatically gives a raise of 3% per year. Write a recursive definition for the geometric sequence formed by the salary increase?

To solve this, we are going to use the recursive formula for a geometric sequence: [tex]a_{n}=a_{1}r^{n-1}[/tex]
where
[tex]a_{n}[/tex] is the nth term of the geometric sequence.
[tex]a_{1}[/tex] is the first term of the geometric sequence.
[tex]r[/tex] is the common ratio 
[tex]n[/tex] is the position of the term in the sequence. 

We know that the starting salary is $42,000, so [tex]a_{1}=42000[/tex]. Now, to find the common ratio [tex]r[/tex], we need to find the next term in the sequence first:
We know from our problem that the company automatically gives a raise of 3% per year, so the next term in the sequence will be 42000 + 3%(42000) = 42000 + 1260 = 43260. Remember that the common ratio is the current term of the geometric sequence divided by the previous term of the sequence; we know that our current term is 43260 and the previous term is 42000, so:
[tex]r= \frac{43260}{42000} [/tex]
[tex]r=1.03[/tex]
Now we can plug the values in our recursive formula:
[tex]a_{n}=a_{1}r^{n-1}[/tex]
[tex]a_{n}=42000(1.03)^{n-1}[/tex]

We can conclude that the recursive definition for the geometric sequence formed by the salary increase is: [tex]a_{n}=42000(1.03)^{n-1}[/tex]