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 15th 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) = 150 \cdot f(n-1), \text{ for } n \geq 2[/tex]

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

C. [tex]f(1) = 1,900[/tex]
[tex]f(n) = f(n-1) + 700, \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 analyze the transactions that Barry makes each month and find the recursive equation that models his account balance.

1. Barry's Monthly Transactions:
- He deposits [tex]$700 from his paycheck.
- He withdraws $[/tex]150 to buy gas.
- He withdraws [tex]$400 for other expenses.

2. Net Monthly Change:
To find how Barry's account balance changes each month, we calculate the net effect of his transactions:
- Total deposit: $[/tex]700
- Total withdrawals: [tex]$150 (for gas) + $[/tex]400 (for other expenses) = [tex]$550

Therefore, the net monthly change in his account balance is:
\[
700 - 550 = 150
\]

3. Understanding the Recursive Equation:
- The balance at the end of month $ n $ is based on the balance at the end of month $ n-1 $ plus the net monthly change.
- Therefore, the recursive equation is:
\[
f(n) = f(n-1) + 150 \text{ for } n \geq 2
\]

4. Initial Condition:
- We are told that Barry's account balance is $[/tex]1,900 at the end of the 15th month. Therefore, we can start from that point:
[tex]\[
f(1) = 1,900
\][/tex]

By following this logic, the correct recursive equation that models Barry's account balance is given by:

Option B:
- [tex]$ f(1) = 1,900 $[/tex]
- [tex]$ f(n) = f(n-1) + 150, \text{ for } n \geq 2 $[/tex]