Answer :
To solve this problem, we need to determine the recursive equation that models Barry's account balance. Let's break down the monthly transactions and their effects:
1. Monthly transactions:
- Barry deposits [tex]$700 into his account from his paycheck.
- Barry withdraws $[/tex]150 for gas.
- Barry withdraws [tex]$400 for other expenses.
2. Calculate the net change in balance each month:
- Monthly deposit = $[/tex]700
- Total monthly withdrawals = [tex]$150 + $[/tex]400 = [tex]$550
- Net change each month = Deposit - Withdrawals = $[/tex]700 - [tex]$550 = $[/tex]150
3. Account balance at end of the first month:
- Given: Barry's account balance at the end of the first month is [tex]$1,900.
4. Understanding the options:
We need to find out which recursive option best fits his account balance after each month.
- Option A: Suggests starting balance of $[/tex]1,000, which is incorrect since we know it’s [tex]$1,900.
- Option B: Incorrect initial balance of $[/tex]1,000 and incorrect monthly change calculation.
- Option C: Incorrect recursive formula used.
- Option D: This states the initial balance correctly as [tex]$1,900 and models a different monthly change. However, it's essential to verify if it correctly represents our calculations.
5. Construct the recursive equation:
From the calculations above, we have:
- Initial account balance after the first month: $[/tex]1,900
- The change in account balance each month thereafter is an increase of [tex]$150, calculated from the net change of $[/tex]700 - $550.
Therefore, the correct recursive formula is:
[tex]\[
f(n) = f(n-1) + 150, \text{ for } n \geq 2
\][/tex]
This matches the balance pattern identified, confirming that the correct recursive equation accurately captures Barry's account ongoing financial transactions.
1. Monthly transactions:
- Barry deposits [tex]$700 into his account from his paycheck.
- Barry withdraws $[/tex]150 for gas.
- Barry withdraws [tex]$400 for other expenses.
2. Calculate the net change in balance each month:
- Monthly deposit = $[/tex]700
- Total monthly withdrawals = [tex]$150 + $[/tex]400 = [tex]$550
- Net change each month = Deposit - Withdrawals = $[/tex]700 - [tex]$550 = $[/tex]150
3. Account balance at end of the first month:
- Given: Barry's account balance at the end of the first month is [tex]$1,900.
4. Understanding the options:
We need to find out which recursive option best fits his account balance after each month.
- Option A: Suggests starting balance of $[/tex]1,000, which is incorrect since we know it’s [tex]$1,900.
- Option B: Incorrect initial balance of $[/tex]1,000 and incorrect monthly change calculation.
- Option C: Incorrect recursive formula used.
- Option D: This states the initial balance correctly as [tex]$1,900 and models a different monthly change. However, it's essential to verify if it correctly represents our calculations.
5. Construct the recursive equation:
From the calculations above, we have:
- Initial account balance after the first month: $[/tex]1,900
- The change in account balance each month thereafter is an increase of [tex]$150, calculated from the net change of $[/tex]700 - $550.
Therefore, the correct recursive formula is:
[tex]\[
f(n) = f(n-1) + 150, \text{ for } n \geq 2
\][/tex]
This matches the balance pattern identified, confirming that the correct recursive equation accurately captures Barry's account ongoing financial transactions.
