Answer :
To solve this problem, we need to model Barry's checking account balance using a recursive equation based on the transactions he makes each month.
Here's a step-by-step explanation:
1. Initial Account Balance:
- At the end of the 1st month, Barry's account balance is [tex]$1,900.
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 out how much Barry's balance changes each month, it's important to calculate the net effect of these transactions.
- Total Deposits: [tex]$700
- Total Withdrawals: $[/tex]150 (for gas) + [tex]$400 (for other expenses) = $[/tex]550
- Net Change in Balance: [tex]$700 (deposits) - $[/tex]550 (withdrawals) = $150
4. Recursive Equation:
- We start with an initial balance at the end of the 1st month: [tex]\( f(1) = 1,900 \)[/tex].
- For subsequent months [tex]\( n \geq 2 \)[/tex], the balance can be calculated using the previous month's balance (since it increases by the net change):
[tex]\[
f(n) = f(n-1) + 150
\][/tex]
By reviewing these calculations, we match them to the options given. The correct recursive equation that models Barry's account balance at the end of month [tex]\( n \)[/tex] is:
A.
[tex]\[ f(1) = 1,900 \][/tex]
[tex]\[ f(n) = f(n-1) + 150, \text{ for } n \geq 2 \][/tex]
Here's a step-by-step explanation:
1. Initial Account Balance:
- At the end of the 1st month, Barry's account balance is [tex]$1,900.
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 out how much Barry's balance changes each month, it's important to calculate the net effect of these transactions.
- Total Deposits: [tex]$700
- Total Withdrawals: $[/tex]150 (for gas) + [tex]$400 (for other expenses) = $[/tex]550
- Net Change in Balance: [tex]$700 (deposits) - $[/tex]550 (withdrawals) = $150
4. Recursive Equation:
- We start with an initial balance at the end of the 1st month: [tex]\( f(1) = 1,900 \)[/tex].
- For subsequent months [tex]\( n \geq 2 \)[/tex], the balance can be calculated using the previous month's balance (since it increases by the net change):
[tex]\[
f(n) = f(n-1) + 150
\][/tex]
By reviewing these calculations, we match them to the options given. The correct recursive equation that models Barry's account balance at the end of month [tex]\( n \)[/tex] is:
A.
[tex]\[ f(1) = 1,900 \][/tex]
[tex]\[ f(n) = f(n-1) + 150, \text{ for } n \geq 2 \][/tex]
