Answer :
Let's solve the problem step-by-step:
1. Understand Monthly Transactions:
- Barry deposits [tex]$700 each month.
- He withdraws $[/tex]150 for gas.
- He withdraws [tex]$400 for other expenses.
2. Calculate the Net Change Each Month:
- Total withdrawals each month = $[/tex]150 (gas) + [tex]$400 (other expenses) = $[/tex]550.
- Net gain or loss each month = [tex]$700 (deposit) - $[/tex]550 (withdrawals) = [tex]$150.
3. Initial Account Balance:
- At the end of the 1st month, Barry's account balance is $[/tex]1,900.
4. Recursive Equation:
- The recursive equation needs to represent his account balance at the end of month [tex]\( n \)[/tex].
- Starting balance is [tex]\( f(1) = 1,900 \)[/tex].
- Since Barry's account increases by $150 each month, the equation for months 2 and above would be the previous month's balance plus the net monthly change:
[tex]\( f(n) = f(n-1) + 150 \)[/tex], for [tex]\( n \geq 2 \)[/tex].
5. Selection of the Correct Answer:
- From the provided options, the equation that matches the above model is:
[tex]\[
D. \quad f(1) = 1,900, \quad f(n) = f(n-1) + 150, \text{ for } n \geq 2
\][/tex]
Therefore, the correct answer is option D.
1. Understand Monthly Transactions:
- Barry deposits [tex]$700 each month.
- He withdraws $[/tex]150 for gas.
- He withdraws [tex]$400 for other expenses.
2. Calculate the Net Change Each Month:
- Total withdrawals each month = $[/tex]150 (gas) + [tex]$400 (other expenses) = $[/tex]550.
- Net gain or loss each month = [tex]$700 (deposit) - $[/tex]550 (withdrawals) = [tex]$150.
3. Initial Account Balance:
- At the end of the 1st month, Barry's account balance is $[/tex]1,900.
4. Recursive Equation:
- The recursive equation needs to represent his account balance at the end of month [tex]\( n \)[/tex].
- Starting balance is [tex]\( f(1) = 1,900 \)[/tex].
- Since Barry's account increases by $150 each month, the equation for months 2 and above would be the previous month's balance plus the net monthly change:
[tex]\( f(n) = f(n-1) + 150 \)[/tex], for [tex]\( n \geq 2 \)[/tex].
5. Selection of the Correct Answer:
- From the provided options, the equation that matches the above model is:
[tex]\[
D. \quad f(1) = 1,900, \quad f(n) = f(n-1) + 150, \text{ for } n \geq 2
\][/tex]
Therefore, the correct answer is option D.