Answer :
Let's solve this step-by-step to find the correct recursive equation for Barry's account balance at the end of month [tex]\( n \)[/tex].
1. Initial Balance:
- At the end of the 1st month, Barry's account balance is [tex]$1,900.
- This can be modeled with \( f(1) = 1,900 \).
2. Monthly Transactions:
- Barry deposits $[/tex]700 from his paycheck.
- He withdraws [tex]$150 to buy gas.
- He withdraws another $[/tex]400 for other expenses.
3. Net Monthly Change:
- To find the net change in his account each month, we calculate the total money coming in and going out:
- Total deposits: [tex]$700
- Total withdrawals: $[/tex]150 (gas) + [tex]$400 (other expenses) = $[/tex]550
- Net monthly change: [tex]$700 (deposits) - $[/tex]550 (withdrawals) = [tex]$150
4. Recursive Equation:
- Each month, the balance increases by this net change of $[/tex]150.
- Therefore, a recursive equation to model this is:
[tex]\[
f(n) = f(n-1) + 150, \text{ for } n \geq 2
\][/tex]
- This equation means that to get the balance for month [tex]\( n \)[/tex], you take the balance from the previous month [tex]\( f(n-1) \)[/tex] and add [tex]$150 to it.
5. Correct Answer:
- Given the choices:
- (A) and (B) suggest different additions or subtractions.
- (C) involves multiplication, which is not the correct operation.
- (D) suggests a \( -150 \), which doesn’t match our calculated net change.
- None of the listed options directly match our derived equation, but option A has a similar format, indicating a technical discrepancy in options. In a typical setting, ensuring the correct equation structure and understanding would be important.
In conclusion, based on the calculations, if correctly recognizing $[/tex]150 as the net change: the closest initial intention was:
[tex]\[
f(n) = f(n-1) + 150, \text{ for } n \geq 2
\][/tex]
1. Initial Balance:
- At the end of the 1st month, Barry's account balance is [tex]$1,900.
- This can be modeled with \( f(1) = 1,900 \).
2. Monthly Transactions:
- Barry deposits $[/tex]700 from his paycheck.
- He withdraws [tex]$150 to buy gas.
- He withdraws another $[/tex]400 for other expenses.
3. Net Monthly Change:
- To find the net change in his account each month, we calculate the total money coming in and going out:
- Total deposits: [tex]$700
- Total withdrawals: $[/tex]150 (gas) + [tex]$400 (other expenses) = $[/tex]550
- Net monthly change: [tex]$700 (deposits) - $[/tex]550 (withdrawals) = [tex]$150
4. Recursive Equation:
- Each month, the balance increases by this net change of $[/tex]150.
- Therefore, a recursive equation to model this is:
[tex]\[
f(n) = f(n-1) + 150, \text{ for } n \geq 2
\][/tex]
- This equation means that to get the balance for month [tex]\( n \)[/tex], you take the balance from the previous month [tex]\( f(n-1) \)[/tex] and add [tex]$150 to it.
5. Correct Answer:
- Given the choices:
- (A) and (B) suggest different additions or subtractions.
- (C) involves multiplication, which is not the correct operation.
- (D) suggests a \( -150 \), which doesn’t match our calculated net change.
- None of the listed options directly match our derived equation, but option A has a similar format, indicating a technical discrepancy in options. In a typical setting, ensuring the correct equation structure and understanding would be important.
In conclusion, based on the calculations, if correctly recognizing $[/tex]150 as the net change: the closest initial intention was:
[tex]\[
f(n) = f(n-1) + 150, \text{ for } n \geq 2
\][/tex]
