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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[/tex], for [tex]n \geq 2[/tex]

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

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

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

Answer :

Let's break down the situation with Barry's checking account to find the correct recursive equation.

Each month, Barry makes the following transactions:

1. Deposit from paycheck: [tex]$+700$[/tex]
2. Withdraw for gas: [tex]$-150$[/tex]
3. Withdraw for other expenses: [tex]$-400$[/tex]

To determine the net change in Barry’s account each month, we need to calculate the total impact of these transactions:

[tex]\[ \text{Net monthly change} = 700 - 150 - 400 \][/tex]

Let's do the math:

- [tex]$700 - 150 = 550$[/tex]
- [tex]$550 - 400 = 150$[/tex]

So, the net change in Barry's account each month is [tex]$+150$[/tex].

Now, let's look at the situation over time:

- Initial balance at the end of the 1st month: [tex]$1,900$[/tex]

Based on the transactions, Barry's balance increases by [tex]$150$[/tex] each month. Therefore, we can write the recursive equation as follows:

- Initial condition: [tex]\( f(1) = 1900 \)[/tex]
- Recursive formula for subsequent months (for [tex]\( n \geq 2 \)[/tex]): [tex]\( f(n) = f(n-1) + 150 \)[/tex]

This means each month Barry's account balance will increase by [tex]$150$[/tex] from the previous month's balance.

Given these steps, the correct answer is choice A:

[tex]\[ f(1) = 1900 \][/tex]
[tex]\[ f(n) = f(n-1) + 150 \quad \text{for} \, n \geq 2 \][/tex]

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