Answer :
To solve the problem, let's carefully analyze Barry's transactions and how they affect his checking account balance each month.
1. Initial Account Balance:
- Barry starts with an account balance of [tex]$1,900 at the end of the first month.
2. Monthly Transactions:
- He deposits $[/tex]700 from his paycheck.
- He withdraws [tex]$150 for gas.
- He withdraws $[/tex]400 for other expenses.
3. Net Change Calculation:
- Each month, Barry deposits and withdraws money. To find the net change in his account balance each month, we calculate:
[tex]\[
\text{Net Change} = \text{Deposit} - (\text{Withdrawal for Gas} + \text{Withdrawal for Other Expenses})
\][/tex]
- Plugging in the values:
[tex]\[
\text{Net Change} = 700 - (150 + 400) = 700 - 550 = 150
\][/tex]
- This means that each month, Barry's account balance increases by [tex]$150 due to his transactions.
4. Recursive Equation:
- Since we know the initial balance and the net change each month, we can write a recursive equation.
- The balance at the end of month 1 is $[/tex]1,900.
- For every subsequent month [tex]\( n \)[/tex], Barry's account balance is his previous month's balance plus the net change:
[tex]\[
f(n) = f(n-1) + 150 \quad \text{for } n \geq 2
\][/tex]
5. Correct Answer:
- The recursive equation modeling Barry's account balance would be:
[tex]\[
f(1) = 1,900
\][/tex]
[tex]\[
f(n) = f(n-1) + 150, \text{ for } n \geq 2
\][/tex]
- Therefore, option A is the correct answer.
This step-by-step process provides a clear understanding of how Barry's account balance changes each month and how the recursive equation is formed.
1. Initial Account Balance:
- Barry starts with an account balance of [tex]$1,900 at the end of the first month.
2. Monthly Transactions:
- He deposits $[/tex]700 from his paycheck.
- He withdraws [tex]$150 for gas.
- He withdraws $[/tex]400 for other expenses.
3. Net Change Calculation:
- Each month, Barry deposits and withdraws money. To find the net change in his account balance each month, we calculate:
[tex]\[
\text{Net Change} = \text{Deposit} - (\text{Withdrawal for Gas} + \text{Withdrawal for Other Expenses})
\][/tex]
- Plugging in the values:
[tex]\[
\text{Net Change} = 700 - (150 + 400) = 700 - 550 = 150
\][/tex]
- This means that each month, Barry's account balance increases by [tex]$150 due to his transactions.
4. Recursive Equation:
- Since we know the initial balance and the net change each month, we can write a recursive equation.
- The balance at the end of month 1 is $[/tex]1,900.
- For every subsequent month [tex]\( n \)[/tex], Barry's account balance is his previous month's balance plus the net change:
[tex]\[
f(n) = f(n-1) + 150 \quad \text{for } n \geq 2
\][/tex]
5. Correct Answer:
- The recursive equation modeling Barry's account balance would be:
[tex]\[
f(1) = 1,900
\][/tex]
[tex]\[
f(n) = f(n-1) + 150, \text{ for } n \geq 2
\][/tex]
- Therefore, option A is the correct answer.
This step-by-step process provides a clear understanding of how Barry's account balance changes each month and how the recursive equation is formed.
