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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]

Answer :

To solve the problem, let's carefully analyze Barry's transactions and how they affect his checking account balance each month.

1. Initial Account Balance:
- Barry starts with an account balance of [tex]$1,900 at the end of the first month.

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

3. Net Change Calculation:
- Each month, Barry deposits and withdraws money. To find the net change in his account balance each month, we calculate:

[tex]\[
\text{Net Change} = \text{Deposit} - (\text{Withdrawal for Gas} + \text{Withdrawal for Other Expenses})
\][/tex]

- Plugging in the values:

[tex]\[
\text{Net Change} = 700 - (150 + 400) = 700 - 550 = 150
\][/tex]

- This means that each month, Barry's account balance increases by [tex]$150 due to his transactions.

4. Recursive Equation:
- Since we know the initial balance and the net change each month, we can write a recursive equation.
- The balance at the end of month 1 is $[/tex]1,900.
- For every subsequent month [tex]\( n \)[/tex], Barry's account balance is his previous month's balance plus the net change:

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

5. Correct Answer:
- The recursive equation modeling Barry's account balance would be:
[tex]\[
f(1) = 1,900
\][/tex]
[tex]\[
f(n) = f(n-1) + 150, \text{ for } n \geq 2
\][/tex]

- Therefore, option A is the correct answer.

This step-by-step process provides a clear understanding of how Barry's account balance changes each month and how the recursive equation is formed.

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