Answer :
To solve the problem, let's analyze the transactions Barry makes each month in his checking account:
1. Barry deposits [tex]$700 from his paycheck.
2. He withdraws $[/tex]150 for gas.
3. He withdraws [tex]$400 for other expenses.
We need to model Barry's account balance at the end of month \( n \) using a recursive equation. Let's break down the steps:
- Initial Balance: Barry's account balance at the end of the first month, \( f(1) \), is given as \( \$[/tex]1,900 \).
- Monthly Change in Balance:
- First, calculate the total withdrawals each month:
- Gas: [tex]\( \$150 \)[/tex]
- Other expenses: [tex]\( \$400 \)[/tex]
- Total withdrawals = [tex]\( 150 + 400 = \$550 \)[/tex]
- Calculate the net effect on the account balance:
[tex]\[
\text{Net change} = \text{Deposit} - \text{Total Withdrawals} = 700 - 550 = \$150
\][/tex]
Thus, each month, Barry's account balance increases by [tex]\( \$150 \)[/tex].
- Recursive Equation:
- For month [tex]\( n \geq 2 \)[/tex], the balance can be expressed as:
[tex]\[
f(n) = f(n-1) + 150
\][/tex]
Now let's review the options:
- Option A: This option suggests multiplying the balance, which isn't correct based on the operations (additions and subtractions) described in the problem.
- Option B: This matches our findings, where each month's balance is the previous month's balance plus [tex]\( \$150 \)[/tex].
- Option C: Incorrect, as it suggests adding the deposit amount without accounting for withdrawals.
- Option D: Incorrect, since it only accounts for the gas withdrawal.
Therefore, the correct answer is:
B. [tex]\( f(1) = 1,900 \)[/tex]; [tex]\( f(n) = f(n-1) + 150 \)[/tex], for [tex]\( n \geq 2 \)[/tex].
1. Barry deposits [tex]$700 from his paycheck.
2. He withdraws $[/tex]150 for gas.
3. He withdraws [tex]$400 for other expenses.
We need to model Barry's account balance at the end of month \( n \) using a recursive equation. Let's break down the steps:
- Initial Balance: Barry's account balance at the end of the first month, \( f(1) \), is given as \( \$[/tex]1,900 \).
- Monthly Change in Balance:
- First, calculate the total withdrawals each month:
- Gas: [tex]\( \$150 \)[/tex]
- Other expenses: [tex]\( \$400 \)[/tex]
- Total withdrawals = [tex]\( 150 + 400 = \$550 \)[/tex]
- Calculate the net effect on the account balance:
[tex]\[
\text{Net change} = \text{Deposit} - \text{Total Withdrawals} = 700 - 550 = \$150
\][/tex]
Thus, each month, Barry's account balance increases by [tex]\( \$150 \)[/tex].
- Recursive Equation:
- For month [tex]\( n \geq 2 \)[/tex], the balance can be expressed as:
[tex]\[
f(n) = f(n-1) + 150
\][/tex]
Now let's review the options:
- Option A: This option suggests multiplying the balance, which isn't correct based on the operations (additions and subtractions) described in the problem.
- Option B: This matches our findings, where each month's balance is the previous month's balance plus [tex]\( \$150 \)[/tex].
- Option C: Incorrect, as it suggests adding the deposit amount without accounting for withdrawals.
- Option D: Incorrect, since it only accounts for the gas withdrawal.
Therefore, the correct answer is:
B. [tex]\( f(1) = 1,900 \)[/tex]; [tex]\( f(n) = f(n-1) + 150 \)[/tex], for [tex]\( n \geq 2 \)[/tex].
