Select the correct answer.

Each month, Barry makes three transactions in his checking account:
- He deposits [tex]$\$700$[/tex] from his paycheck.
- He withdraws [tex]$\$150$[/tex] to buy gas for his car.
- He withdraws [tex]$\$400$[/tex] for other expenses.

If his account balance is [tex]$\$1,900$[/tex] at the end of the 1st month, which recursive equation models Barry's account balance at the end of month [tex]$n$[/tex]?

A. [tex]f(1)=1,900[/tex]
[tex]f(n)=f(n-1)+150, \text{ for } n \geq 2[/tex]

B. [tex]f(1)=1,900[/tex]
[tex]f(n)=f(n-1)+700, \text{ for } n \geq 2[/tex]

C. [tex]f(1)=1,900[/tex]
[tex]f(n)=150 \cdot f(n-1), \text{ for } n \geq 2[/tex]

D. [tex]f(1)=1,900[/tex]
[tex]f(n)=f(n-1)-150, \text{ for } n \geq 2[/tex]

Let's solve the problem step-by-step:

1. Understand Monthly Transactions:
- Barry deposits [tex]$700 each month.
- He withdraws $[/tex]150 for gas.
- He withdraws [tex]$400 for other expenses.

2. Calculate the Net Change Each Month:
- Total withdrawals each month = $[/tex]150 (gas) + [tex]$400 (other expenses) = $[/tex]550.
- Net gain or loss each month = [tex]$700 (deposit) - $[/tex]550 (withdrawals) = [tex]$150.

3. Initial Account Balance:
- At the end of the 1st month, Barry's account balance is $[/tex]1,900.

4. Recursive Equation:
- The recursive equation needs to represent his account balance at the end of month [tex]$ n $[/tex].
- Starting balance is [tex]$ f(1) = 1,900 $[/tex].
- Since Barry's account increases by $150 each month, the equation for months 2 and above would be the previous month's balance plus the net monthly change:
[tex]$ f(n) = f(n-1) + 150 $[/tex], for [tex]$ n \geq 2 $[/tex].

5. Selection of the Correct Answer:
- From the provided options, the equation that matches the above model is:
[tex]\[
D. \quad f(1) = 1,900, \quad f(n) = f(n-1) + 150, \text{ for } n \geq 2
\][/tex]

Therefore, the correct answer is option D.