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.

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 :

Barry's account balance changes each month based on three transactions:

1. He deposits [tex]$\$[/tex]700[tex]$.
2. He withdraws $[/tex]\[tex]$150$[/tex] for gas.
3. He withdraws [tex]$\$[/tex]400[tex]$ for other expenses.

First, we calculate the total amount withdrawn each month:
$[/tex][tex]$
\text{Total withdrawal} = 150 + 400 = 550.
$[/tex][tex]$

Next, we determine the net monthly change by subtracting the total withdrawals from the deposit:
$[/tex][tex]$
\text{Net change} = 700 - 550 = 150.
$[/tex][tex]$

This means that every month, Barry's account increases by $[/tex]\[tex]$150$[/tex]. Given that his account balance at the end of month 1 is [tex]$\$[/tex]1900[tex]$, the recursive formula for his account balance at the end of month $[/tex]n[tex]$ is:
$[/tex][tex]$
\begin{aligned}
f(1) &= 1900, \\
f(n) &= f(n-1) + 150 \quad \text{for } n \ge 2.
\end{aligned}
$[/tex]$

Thus, the correct option is D.

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