High School

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) = 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 solve this problem, we need to determine how Barry's account balance changes monthly based on his transactions and select the correct recursive equation. Let's break it down step-by-step:

1. Transactions Details:
- Deposit: Barry deposits [tex]$700 each month from his paycheck.
- Withdrawals:
- He withdraws $[/tex]150 for gas.
- He withdraws [tex]$400 for other expenses.

2. Net Change Calculation:
- Each month's net change in balance is calculated by subtracting the total withdrawals from the deposit:
\[
\text{Net Change} = 700 - (150 + 400) = 700 - 550 = 150
\]
- Each month, Barry’s account increases by $[/tex]150 after all his transactions.

3. Modeling the Recursive Equation:
- At the end of the 1st month, his account balance is [tex]$1,900.
- Starting from the 2nd month, his balance at the end of each month increases by $[/tex]150.

4. Formulating the Recursive Equation:
- We use "f(n)" to denote Barry's account balance at the end of month [tex]\( n \)[/tex].
- The initial condition is:
[tex]\[
f(1) = 1,900
\][/tex]
- For each subsequent month [tex]\( n \geq 2 \)[/tex], the balance can be described recursively as:
[tex]\[
f(n) = f(n-1) + 150
\][/tex]

5. Choosing the Correct Option:
- Based on our analysis, the equation that models Barry's account balance correctly is:
- Initial balance: [tex]\( f(1) = 1,900 \)[/tex]
- Recursive relation: [tex]\( f(n) = f(n-1) + 150 \)[/tex]

So, the correct answer is B:
[tex]\[ f(1) = 1,900 \][/tex]
[tex]\[ f(n) = f(n-1) + 150, \text{ for } n \geq 2 \][/tex]