Question

Salary Increases A newly hired salesman is promised a beginning salary of$\backslash (30,000$a year with a$\backslash )2000$raise every year. Let$S_n$be his salary in histh year of employment.
(a) Find a recursive definition of $S_n$.
(b) Find his salary in his fifth year of employment.

Short answer

(a). A newly hired salesman is promised a beginning salary of $\$30,000$ a year with a $\$$2000 raise every year.

So the recursive definition of$S_n$is$S_{0_{}}=30000$ and $S_n=S_{n-1}+2000$ .

(b). The salary of the salesman in his fifth year of employment is $\$\text{38,000}$.

Step-by-step solution

a.Step 1.Given information.

Given that the initial salary of a salesman is $\$30,000$ and their is a raise of $\$$2000 every year.

Step 2. The objective is to find the recursive definition of Sn where Sn is the salary of the salesman in the nth year of his employment.

The initial salary$S_{0_{}}=30000$ and the salary increases by 2000 dollars every year. So the salary in the nth year of employment will be 2000 more than the salary in the(n-1)th year. Thus $S_n=S_{n-1}+2000$

As the n th term depends on the previous terms hence the sequence is recursive.

b.Step 1.Given information.

The recursive definition of $S_n$ is $S_{0_{}}=30000$ and $S_n=S_{n-1}+2000$

Step 2:The objective is to find the salary of the salesman in his 5 th year of employment.

Put the values in equation then solve for the required salary

The initial salary $S_{0_{}}$is $\$30,000$ and the salary in the nth year of employment is$S_n=S_{n-1}+2000$

To find the salary in the fifth year of employment we need to calculate${\text{S}}_4$

For n=1,

$\begin{array}{c}{S}_1=S_0+2000\\ =30000+2000\\ =32000\end{array}$

For n=2,

$\begin{array}{c}{S}_2=S_1+2000\\ =32000+2000\\ =34000\end{array}$

For n=3,

$\begin{array}{l}{S}_3=S_2+2000\\ =34000+2000\\ =36000\end{array}$

For n=4,

$\begin{array}{l}{S}_4=S_3+2000\\ =36000+2000\\ =38000\end{array}$