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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]$f(n) = f(n-1) + 700$[/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]$f(n) = 150 \cdot f(n-1)$[/tex], for [tex]$n \geq 2$[/tex]

Answer :

Let's break down the problem and find the recursive equation step-by-step:

1. Understand Barry’s Monthly Transactions:
- Deposit: Barry deposits [tex]$700 each month.
- Withdrawals: Barry withdraws $[/tex]150 for gas and [tex]$400 for other expenses, totaling $[/tex]150 + [tex]$400 = $[/tex]550 per month.

2. Calculate Net Change Per Month:
- The net change in Barry’s account each month is calculated by subtracting the withdrawals from the deposits:
[tex]\[
\text{Net change} = \$700 - \$550 = \$150
\][/tex]

3. Initial Account Balance:
- At the end of the 1st month, Barry’s account balance is given as [tex]$1,900.

4. Set Up the Recursive Equation:
- We know the balance at the end of the 1st month is $[/tex]1,900, so:
[tex]\[
f(1) = 1,900
\][/tex]
- For each subsequent month [tex]\( n \geq 2 \)[/tex], the account balance is equal to the previous month’s balance plus the net change of $150:
[tex]\[
f(n) = f(n-1) + 150
\][/tex]

Based on these observations, the correct recursive equation is:
- [tex]\( f(1) = 1,900 \)[/tex]
- [tex]\( f(n) = f(n-1) + 150 \)[/tex], for [tex]\( n \geq 2 \)[/tex]

Thus, the correct answer is:

A. [tex]\( f(1) = 1,900 \)[/tex]
[tex]\( f(n) = f(n-1) + 150, \text{ for } n \geq 2 \)[/tex]

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