Let's analyze the problem step-by-step to determine the correct answer for the recursive equation modeling Barry's account balance.
1. Initial Balance:
- Barry's starting balance at the end of the first month is \[tex]$1,900.
- Therefore, at the end of month 1, the account balance \( f(1) \) is \$[/tex]1,900.
- Thus, one component of the recursive equation is [tex]\( f(1) = 1,900 \)[/tex].
2. Monthly Transactions:
- Barry deposits \[tex]$700 from his paycheck.
- He withdraws \$[/tex]150 for gas.
- He withdraws \[tex]$400 for other expenses.
3. Net Monthly Change:
- To find the net change in the balance each month, calculate the total withdrawals:
\[
\text{Total withdrawals} = \text{withdrawal for gas} + \text{withdrawal for other expenses} = 150 + 400 = 550 \text{ dollars}
\]
- The net change per month would be:
\[
\text{Net change} = \text{deposit} - \text{total withdrawals} = 700 - 550 = 150 \text{ dollars}
\]
- This means that each month, Barry's balance increases by \$[/tex]150.
4. Recursive Equation:
- The recursive equation for the balance at the end of month [tex]\( n \)[/tex] can be expressed as:
[tex]\[
f(n) = f(n-1) + 150, \quad \text{for } n \geq 2
\][/tex]
- This satisfies the condition that each month the account balance increases by \$150 due to the net change after deposits and withdrawals.
Given the choices, the correct answer is:
D.
[tex]\[
\begin{array}{l}
f(1) = 1,900 \\
f(n) = f(n-1) + 150, \text{ for } n \geq 2
\end{array}
\][/tex]
This matches the correct net monthly change and indicates the growth in the account balance accurately.
