Answer :
To solve this problem, let's break down Barry's monthly transactions and analyze how they affect his account balance, and then determine which recursive equation best models his balance at the end of each month.
1. Understand Barry's Transactions:
- Each month, Barry performs the following:
- He deposits [tex]$700 from his paycheck.
- He withdraws $[/tex]150 to buy gas.
- He withdraws [tex]$400 for other expenses.
2. Calculate the Net Change for the Month:
- To find the net effect on his account balance, subtract the total withdrawals from the deposit:
\[
\text{Net change} = \$[/tex]700 - (\[tex]$150 + \$[/tex]400)
\]
- Performing the calculation:
[tex]\[
\text{Net change} = \$700 - \$550 = \$150
\][/tex]
3. Determine the Recursive Equation:
- Since Barry's balance changes by a net amount of [tex]$150 each month, we need to determine how his balance at the end of month \( n \) relates to the balance at the end of month \( n-1 \).
- The balance for month 1 is given as $[/tex]1,900.
4. Formulate the Recursive Equation:
- The recursive equation is constructed as:
- [tex]\( f(1) = 1,900 \)[/tex]
- For [tex]\( n \geq 2 \)[/tex], [tex]\( f(n) = f(n-1) + 150 \)[/tex]
5. Match with Given Options:
- Comparing with the given choices, option D, which states:
- [tex]\( f(1) = 1,900 \)[/tex]
- [tex]\( f(n) = f(n-1) + 150 \)[/tex], for [tex]\( n \geq 2 \)[/tex]
- This matches the computed result, thus option D is the correct recursive equation to model Barry's account balance at the end of month [tex]\( n \)[/tex].
Therefore, the correct answer is option D.
1. Understand Barry's Transactions:
- Each month, Barry performs the following:
- He deposits [tex]$700 from his paycheck.
- He withdraws $[/tex]150 to buy gas.
- He withdraws [tex]$400 for other expenses.
2. Calculate the Net Change for the Month:
- To find the net effect on his account balance, subtract the total withdrawals from the deposit:
\[
\text{Net change} = \$[/tex]700 - (\[tex]$150 + \$[/tex]400)
\]
- Performing the calculation:
[tex]\[
\text{Net change} = \$700 - \$550 = \$150
\][/tex]
3. Determine the Recursive Equation:
- Since Barry's balance changes by a net amount of [tex]$150 each month, we need to determine how his balance at the end of month \( n \) relates to the balance at the end of month \( n-1 \).
- The balance for month 1 is given as $[/tex]1,900.
4. Formulate the Recursive Equation:
- The recursive equation is constructed as:
- [tex]\( f(1) = 1,900 \)[/tex]
- For [tex]\( n \geq 2 \)[/tex], [tex]\( f(n) = f(n-1) + 150 \)[/tex]
5. Match with Given Options:
- Comparing with the given choices, option D, which states:
- [tex]\( f(1) = 1,900 \)[/tex]
- [tex]\( f(n) = f(n-1) + 150 \)[/tex], for [tex]\( n \geq 2 \)[/tex]
- This matches the computed result, thus option D is the correct recursive equation to model Barry's account balance at the end of month [tex]\( n \)[/tex].
Therefore, the correct answer is option D.
