High School

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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.

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]$m$[/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)=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)+700[/tex], for [tex]n \geq 2[/tex]

Answer :

Let's solve the problem step by step to determine the correct recursive equation for Barry's account balance.

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

2. Calculate the Net Change Each Month:
- The total amount Barry withdraws each month is \$[/tex]150 (gas) + \[tex]$400 (expenses) = \$[/tex]550.
- The difference between his deposit and his withdrawals is \[tex]$700 (deposit) - \$[/tex]550 (total withdrawals) = \[tex]$150.

3. Initial Condition:
- At the end of the 1st month, Barry's account balance is given as \$[/tex]1,900.

4. Recursive Equation:
- Since each month, Barry's balance increases by the net change amount, the recursive equation can be written as:
- [tex]\( f(1) = 1,900 \)[/tex]
- [tex]\( f(n) = f(n-1) + 150 \)[/tex], for [tex]\( n \geq 2 \)[/tex]

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