Answer :
To solve this problem, we're looking for a recursive equation that models Barry's account balance at the end of month [tex]\( m \)[/tex].
1. Understand the Transactions:
- Barry deposits \[tex]$700 each month.
- Barry withdraws \$[/tex]150 for gas.
- Barry also withdraws \[tex]$400 for other expenses.
2. Calculate the Net Monthly Change:
To find out how Barry's account balance changes each month, we calculate the total withdrawals and subtract it from the deposit:
\[
\text{Net monthly change} = \$[/tex]700 - (\[tex]$150 + \$[/tex]400) = \[tex]$700 - \$[/tex]550 = \[tex]$150
\]
This means each month, Barry adds \$[/tex]150 to his account after all transactions.
3. Define the Recursive Equation:
We are given that Barry's account balance at the end of the first month is \[tex]$1,900. So we define:
\[
f(1) = 1900
\]
For each subsequent month \( n \geq 2 \), the account balance is the previous month’s balance plus the net change of \$[/tex]150:
[tex]\[
f(n) = f(n-1) + 150, \text{ for } n \geq 2
\][/tex]
4. Select the Correct Answer:
Looking at the available choices in the problem, the correct recursive equation that matches our findings is:
[tex]\[
\text{Choice D: } f(1) = 1,900; \, f(n) = f(n-1) + 150, \text{ for } n \geq 2
\][/tex]
Thus, the correct answer is D.
1. Understand the Transactions:
- Barry deposits \[tex]$700 each month.
- Barry withdraws \$[/tex]150 for gas.
- Barry also withdraws \[tex]$400 for other expenses.
2. Calculate the Net Monthly Change:
To find out how Barry's account balance changes each month, we calculate the total withdrawals and subtract it from the deposit:
\[
\text{Net monthly change} = \$[/tex]700 - (\[tex]$150 + \$[/tex]400) = \[tex]$700 - \$[/tex]550 = \[tex]$150
\]
This means each month, Barry adds \$[/tex]150 to his account after all transactions.
3. Define the Recursive Equation:
We are given that Barry's account balance at the end of the first month is \[tex]$1,900. So we define:
\[
f(1) = 1900
\]
For each subsequent month \( n \geq 2 \), the account balance is the previous month’s balance plus the net change of \$[/tex]150:
[tex]\[
f(n) = f(n-1) + 150, \text{ for } n \geq 2
\][/tex]
4. Select the Correct Answer:
Looking at the available choices in the problem, the correct recursive equation that matches our findings is:
[tex]\[
\text{Choice D: } f(1) = 1,900; \, f(n) = f(n-1) + 150, \text{ for } n \geq 2
\][/tex]
Thus, the correct answer is D.
