Question

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?

Answer 1

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

We know that the starting salary is $42,000, so $a_{1}=42000$. Now, to find the common ratio $r$, 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:
$r= \frac{43260}{42000}$
$r=1.03$
Now we can plug the values in our recursive formula:
$a_{n}=a_{1}r^{n-1}$
$a_{n}=42000(1.03)^{n-1}$

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