Answer :
To solve this problem, we need to determine how Barry's account balance changes monthly based on his transactions and select the correct recursive equation. Let's break it down step-by-step:
1. Transactions Details:
- Deposit: Barry deposits [tex]$700 each month from his paycheck.
- Withdrawals:
- He withdraws $[/tex]150 for gas.
- He withdraws [tex]$400 for other expenses.
2. Net Change Calculation:
- Each month's net change in balance is calculated by subtracting the total withdrawals from the deposit:
\[
\text{Net Change} = 700 - (150 + 400) = 700 - 550 = 150
\]
- Each month, Barry’s account increases by $[/tex]150 after all his transactions.
3. Modeling the Recursive Equation:
- At the end of the 1st month, his account balance is [tex]$1,900.
- Starting from the 2nd month, his balance at the end of each month increases by $[/tex]150.
4. Formulating the Recursive Equation:
- We use "f(n)" to denote Barry's account balance at the end of month [tex]\( n \)[/tex].
- The initial condition is:
[tex]\[
f(1) = 1,900
\][/tex]
- For each subsequent month [tex]\( n \geq 2 \)[/tex], the balance can be described recursively as:
[tex]\[
f(n) = f(n-1) + 150
\][/tex]
5. Choosing the Correct Option:
- Based on our analysis, the equation that models Barry's account balance correctly is:
- Initial balance: [tex]\( f(1) = 1,900 \)[/tex]
- Recursive relation: [tex]\( f(n) = f(n-1) + 150 \)[/tex]
So, the correct answer is B:
[tex]\[ f(1) = 1,900 \][/tex]
[tex]\[ f(n) = f(n-1) + 150, \text{ for } n \geq 2 \][/tex]
1. Transactions Details:
- Deposit: Barry deposits [tex]$700 each month from his paycheck.
- Withdrawals:
- He withdraws $[/tex]150 for gas.
- He withdraws [tex]$400 for other expenses.
2. Net Change Calculation:
- Each month's net change in balance is calculated by subtracting the total withdrawals from the deposit:
\[
\text{Net Change} = 700 - (150 + 400) = 700 - 550 = 150
\]
- Each month, Barry’s account increases by $[/tex]150 after all his transactions.
3. Modeling the Recursive Equation:
- At the end of the 1st month, his account balance is [tex]$1,900.
- Starting from the 2nd month, his balance at the end of each month increases by $[/tex]150.
4. Formulating the Recursive Equation:
- We use "f(n)" to denote Barry's account balance at the end of month [tex]\( n \)[/tex].
- The initial condition is:
[tex]\[
f(1) = 1,900
\][/tex]
- For each subsequent month [tex]\( n \geq 2 \)[/tex], the balance can be described recursively as:
[tex]\[
f(n) = f(n-1) + 150
\][/tex]
5. Choosing the Correct Option:
- Based on our analysis, the equation that models Barry's account balance correctly is:
- Initial balance: [tex]\( f(1) = 1,900 \)[/tex]
- Recursive relation: [tex]\( f(n) = f(n-1) + 150 \)[/tex]
So, the correct answer is B:
[tex]\[ f(1) = 1,900 \][/tex]
[tex]\[ f(n) = f(n-1) + 150, \text{ for } n \geq 2 \][/tex]
