Answer :
Let's solve the problem step-by-step to find the correct recursive equation for Barry's account balance.
1. Understand the Transactions:
- Each month, Barry performs the following transactions in his checking account:
- He deposits [tex]$700 from his paycheck.
- He withdraws $[/tex]150 to buy gas for his car.
- He withdraws [tex]$400 for other expenses.
2. Calculate the Net Monthly Change:
- Barry's account balance changes by the net effect of these transactions each month.
- The total deposited amount is $[/tex]700.
- The total withdrawn amount is [tex]$150 (for gas) + $[/tex]400 (for other expenses) = [tex]$550.
- Therefore, the net change every month is: $[/tex]700 - [tex]$550 = $[/tex]150.
3. Initial Account Balance:
- At the end of the first month, Barry's balance is [tex]$1,900.
4. Formulate the Recursive Equation:
- We need to set up a recursive equation for Barry's account balance at the end of month \( n \).
- At the start of month 2 and every subsequent month, Barry's account balance increases by the net change calculated, which is $[/tex]150.
Given these details, the recursive equation to model Barry's balance at the end of month [tex]\( n \)[/tex] is:
- [tex]\( f(1) = 1,900 \)[/tex]
- [tex]\( f(n) = f(n-1) + 150 \)[/tex], for [tex]\( n \geq 2 \)[/tex]
This matches option A in the problem statement, which indicates:
[tex]\[ f(n) = f(n-1) + 150 \][/tex]
However, since none of the options provided initially match our calculated answer, it seems there's a mismatch or error in the given options. The corrected recursive equation is listed above.
1. Understand the Transactions:
- Each month, Barry performs the following transactions in his checking account:
- He deposits [tex]$700 from his paycheck.
- He withdraws $[/tex]150 to buy gas for his car.
- He withdraws [tex]$400 for other expenses.
2. Calculate the Net Monthly Change:
- Barry's account balance changes by the net effect of these transactions each month.
- The total deposited amount is $[/tex]700.
- The total withdrawn amount is [tex]$150 (for gas) + $[/tex]400 (for other expenses) = [tex]$550.
- Therefore, the net change every month is: $[/tex]700 - [tex]$550 = $[/tex]150.
3. Initial Account Balance:
- At the end of the first month, Barry's balance is [tex]$1,900.
4. Formulate the Recursive Equation:
- We need to set up a recursive equation for Barry's account balance at the end of month \( n \).
- At the start of month 2 and every subsequent month, Barry's account balance increases by the net change calculated, which is $[/tex]150.
Given these details, the recursive equation to model Barry's balance at the end of month [tex]\( n \)[/tex] is:
- [tex]\( f(1) = 1,900 \)[/tex]
- [tex]\( f(n) = f(n-1) + 150 \)[/tex], for [tex]\( n \geq 2 \)[/tex]
This matches option A in the problem statement, which indicates:
[tex]\[ f(n) = f(n-1) + 150 \][/tex]
However, since none of the options provided initially match our calculated answer, it seems there's a mismatch or error in the given options. The corrected recursive equation is listed above.
