Answer :
To solve this problem, let's break down Barry's transactions and see how they affect his account balance each month.
Each month, Barry:
1. Deposits [tex]$700 into his account.
2. Withdraws $[/tex]150 for gas.
3. Withdraws [tex]$400 for other expenses.
To find the net change in his account balance each month, we can calculate it by looking at the total deposits and withdrawals:
- Total deposits = $[/tex]700
- Total withdrawals = [tex]$150 (gas) + $[/tex]400 (expenses) = [tex]$550
The net change in the balance each month is therefore:
\[ \text{Net change} = 700 - 550 = 150 \]
Barry's account balance at the end of the first month is $[/tex]1,900. From the second month onward, the balance changes by [tex]$150 each month. We can express this situation as a recursive equation:
1. The initial condition is that at the end of the first month, his balance is $[/tex]1,900:
[tex]\[ f(1) = 1,900 \][/tex]
2. For any month [tex]\( n \geq 2 \)[/tex], the account balance is the previous month's balance plus the net change of [tex]$150:
\[ f(n) = f(n-1) + 150 \]
Therefore, the recursive equation that models Barry's account balance at the end of month \( n \) is:
C. \( f(1) = 1,900 \)
\( f(n) = f(n-1) + 150 \), for \( n \geq 2 \)
This equation accounts for the monthly deposit and withdrawals, reflecting the net increase of $[/tex]150 each month.
Each month, Barry:
1. Deposits [tex]$700 into his account.
2. Withdraws $[/tex]150 for gas.
3. Withdraws [tex]$400 for other expenses.
To find the net change in his account balance each month, we can calculate it by looking at the total deposits and withdrawals:
- Total deposits = $[/tex]700
- Total withdrawals = [tex]$150 (gas) + $[/tex]400 (expenses) = [tex]$550
The net change in the balance each month is therefore:
\[ \text{Net change} = 700 - 550 = 150 \]
Barry's account balance at the end of the first month is $[/tex]1,900. From the second month onward, the balance changes by [tex]$150 each month. We can express this situation as a recursive equation:
1. The initial condition is that at the end of the first month, his balance is $[/tex]1,900:
[tex]\[ f(1) = 1,900 \][/tex]
2. For any month [tex]\( n \geq 2 \)[/tex], the account balance is the previous month's balance plus the net change of [tex]$150:
\[ f(n) = f(n-1) + 150 \]
Therefore, the recursive equation that models Barry's account balance at the end of month \( n \) is:
C. \( f(1) = 1,900 \)
\( f(n) = f(n-1) + 150 \), for \( n \geq 2 \)
This equation accounts for the monthly deposit and withdrawals, reflecting the net increase of $[/tex]150 each month.
