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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)=150 \cdot f(n-1), \text{ for } 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 determine the correct recursive equation for modeling Barry's account balance at the end of each month, let's break down the transactions and calculate the total change to his account balance each month.

### Transactions Each Month:
1. Deposit from paycheck: [tex]$\$[/tex]700[tex]$
2. Withdraw for gas: $[/tex]\[tex]$150$[/tex]
3. Withdraw for other expenses: [tex]$\$[/tex]400[tex]$

### Total Monthly Change:
- The total change in Barry's account each month is calculated by adding his deposit and subtracting his withdrawals:
\[
\text{Total Change} = \text{Deposit} - \text{Withdraw for gas} - \text{Withdraw for other expenses}
\]
\[
\text{Total Change} = 700 - 150 - 400
\]
\[
\text{Total Change} = 150
\]

This means that Barry's account balance increases by $[/tex]\[tex]$150$[/tex] each month.

### Recursive Equation:
- Initially, at the end of the 1st month, Barry's balance is [tex]$\$[/tex]1,900$. This is given as [tex]\( f(1) = 1,900 \)[/tex].
- For each subsequent month [tex]\( n \)[/tex], the balance is the balance of the previous month plus the total monthly change:
[tex]\[
f(n) = f(n-1) + 150 \quad \text{for} \quad n \geq 2
\][/tex]

Thus, the recursive equation that models Barry's account balance at the end of month [tex]\( n \)[/tex] is:

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

This option correctly represents the change in Barry's account balance month over month.

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