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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 first 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]$f(n)=150 \cdot f(n-1)$[/tex], for [tex]$n \geq 2$[/tex]

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

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

D. [tex]$f(1)=1,900$[/tex], [tex]$f(n)=f(n-1)+700$[/tex], for [tex]$n \geq 2$[/tex]

Answer :

To determine which recursive equation models Barry's account balance, let's break down his monthly transactions:

1. Beginning of the 1st Month:
- Initial account balance: [tex]$1,900

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

3. Net Monthly Change Calculation:
- Total withdrawals = [tex]$150 (gas) + $[/tex]400 (other expenses) = [tex]$550
- Net change each month = $[/tex]700 (deposit) - [tex]$550 (total withdrawals) = $[/tex]150

4. Recursive Equation:
- The initial balance at the end of the 1st month is [tex]$1,900.
- Each subsequent month's balance is the previous month's balance plus the net change of $[/tex]150.
- Recursive formula:
- [tex]\( f(1) = 1,900 \)[/tex] (initial balance)
- [tex]\( f(n) = f(n-1) + 150 \)[/tex] for [tex]\( n \geq 2 \)[/tex]

Based on this analysis, the answer is Option B:
[tex]\[ f(1) = 1,900 \][/tex]
[tex]\[ f(n) = f(n-1) + 150, \text{ for } n \geq 2 \][/tex]

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