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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 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]$f(n) = f(n-1) + 700, \text{ for } n \geq 2$[/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]$f(n) = 150 \cdot f(n-1), \text{ for } n \geq 2$[/tex]

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

Answer :

Let's solve this step-by-step to find the correct recursive equation for Barry's account balance at the end of month [tex]\( n \)[/tex].

1. Initial Balance:
- At the end of the 1st month, Barry's account balance is [tex]$1,900.
- This can be modeled with \( f(1) = 1,900 \).

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

3. Net Monthly Change:
- To find the net change in his account each month, we calculate the total money coming in and going out:
- Total deposits: [tex]$700
- Total withdrawals: $[/tex]150 (gas) + [tex]$400 (other expenses) = $[/tex]550
- Net monthly change: [tex]$700 (deposits) - $[/tex]550 (withdrawals) = [tex]$150

4. Recursive Equation:
- Each month, the balance increases by this net change of $[/tex]150.
- Therefore, a recursive equation to model this is:
[tex]\[
f(n) = f(n-1) + 150, \text{ for } n \geq 2
\][/tex]
- This equation means that to get the balance for month [tex]\( n \)[/tex], you take the balance from the previous month [tex]\( f(n-1) \)[/tex] and add [tex]$150 to it.

5. Correct Answer:
- Given the choices:
- (A) and (B) suggest different additions or subtractions.
- (C) involves multiplication, which is not the correct operation.
- (D) suggests a \( -150 \), which doesn’t match our calculated net change.
- None of the listed options directly match our derived equation, but option A has a similar format, indicating a technical discrepancy in options. In a typical setting, ensuring the correct equation structure and understanding would be important.

In conclusion, based on the calculations, if correctly recognizing $[/tex]150 as the net change: the closest initial intention was:
[tex]\[
f(n) = f(n-1) + 150, \text{ for } n \geq 2
\][/tex]

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