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

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

Answer :

Sure! Let's go through the problem step-by-step to determine which recursive equation best models Barry's account balance at the end of each month.

1. Understand Barry's Transactions:
- Each month, Barry deposits [tex]$700 from his paycheck.
- He withdraws $[/tex]150 for gas.
- He withdraws [tex]$400 for other expenses.

2. Calculate the Net Change Each Month:
- The total deposits each month is $[/tex]700.
- The total withdrawals each month are [tex]$150 (for gas) + $[/tex]400 (for other expenses) = [tex]$550.
- Therefore, the net change in his account each month is $[/tex]700 (deposit) - [tex]$550 (withdrawals) = $[/tex]150.

3. Determine the Recursive Equation:
- We know that the initial balance at the end of the 1st month is [tex]$1,900. So, \( f(1) = 1,900 \).
- The monthly net change is positive $[/tex]150, indicating that the balance increases by [tex]$150 each month.
- The recursive equation that captures this monthly change would be \( f(n) = f(n-1) + 150 \) for \( n \geq 2 \).

4. Match with the Given Options:
- Option A: \( f(n) = f(n-1) - 150 \) implies the balance decreases by $[/tex]150 each month.
- Option B: [tex]\( f(n) = f(n-1) + 150 \)[/tex] implies the balance increases by [tex]$150 each month.
- Option C: \( f(n) = f(n-1) + 700 \) implies the balance increases by $[/tex]700 each month.
- Option D: [tex]\( f(n) = 150 \cdot f(n-1) \)[/tex] is a multiplicative relationship, not applicable here.

Therefore, the correct equation is Option B:

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

This equation accurately models the net change in Barry's account balance at the end of each month.

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