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Answer :
To solve this problem, we need to model Barry's account balance using a recursive equation for each month. Here's how we can arrive at the correct equation step-by-step:
1. Determine the Initial Balance:
At the end of the 1st month, Barry's account balance is \[tex]$1,900. This gives us the initial condition:
\[
f(1) = 1900
\]
2. Identify Transactions Each Month:
- Barry deposits \$[/tex]700 from his paycheck.
- He withdraws \[tex]$150 for gas.
- He withdraws \$[/tex]400 for other expenses.
3. Calculate Net Change Each Month:
The net change in Barry's account for each month is calculated as follows:
[tex]\[
\text{Net Change} = \text{Deposit} - (\text{Withdraw for Gas} + \text{Withdraw for Other Expenses})
\][/tex]
Plug in the numbers:
[tex]\[
\text{Net Change} = 700 - (150 + 400) = 700 - 550 = 150
\][/tex]
4. Set Up the Recursive Equation:
The balance at the end of month [tex]\( m \)[/tex] depends on the balance at the previous month [tex]\( (m-1) \)[/tex], plus the net change of \$150:
[tex]\[
f(n) = f(n-1) + 150, \text{ for } n \geq 2
\][/tex]
5. Conclusion:
Barry's account can be modeled with the recursive equations:
[tex]\[
f(1) = 1900
\][/tex]
[tex]\[
f(n) = f(n-1) + 150, \text{ for } n \geq 2
\][/tex]
The correct answer matches option A:
- [tex]\( f(1) = 1900 \)[/tex]
- [tex]\( f(n) = f(n-1) + 150, \text{ for } n \geq 2 \)[/tex]
1. Determine the Initial Balance:
At the end of the 1st month, Barry's account balance is \[tex]$1,900. This gives us the initial condition:
\[
f(1) = 1900
\]
2. Identify Transactions Each Month:
- Barry deposits \$[/tex]700 from his paycheck.
- He withdraws \[tex]$150 for gas.
- He withdraws \$[/tex]400 for other expenses.
3. Calculate Net Change Each Month:
The net change in Barry's account for each month is calculated as follows:
[tex]\[
\text{Net Change} = \text{Deposit} - (\text{Withdraw for Gas} + \text{Withdraw for Other Expenses})
\][/tex]
Plug in the numbers:
[tex]\[
\text{Net Change} = 700 - (150 + 400) = 700 - 550 = 150
\][/tex]
4. Set Up the Recursive Equation:
The balance at the end of month [tex]\( m \)[/tex] depends on the balance at the previous month [tex]\( (m-1) \)[/tex], plus the net change of \$150:
[tex]\[
f(n) = f(n-1) + 150, \text{ for } n \geq 2
\][/tex]
5. Conclusion:
Barry's account can be modeled with the recursive equations:
[tex]\[
f(1) = 1900
\][/tex]
[tex]\[
f(n) = f(n-1) + 150, \text{ for } n \geq 2
\][/tex]
The correct answer matches option A:
- [tex]\( f(1) = 1900 \)[/tex]
- [tex]\( f(n) = f(n-1) + 150, \text{ for } n \geq 2 \)[/tex]
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