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Answer :
- Calculate the net monthly change in Barry's account: $700 (deposit) - $150 (gas) - $400 (other expenses) = $150.
- Define $f(n)$ as the account balance at the end of month $n$, with $f(1) = $1900.
- Express the recursive equation as $f(n) = f(n-1) + $150 for $n \geq 2$, reflecting the $150 increase each month.
- The correct recursive equation is $\boxed{f(1) = 1900, f(n) = f(n-1) + 150}$ for $n \geq 2$.
### Explanation
1. Analyzing Monthly Transactions
Let's analyze Barry's monthly transactions to determine how his account balance changes each month. He deposits $700 and withdraws $150 and $400. We need to find the net change in his account balance each month.
2. Calculating Net Change
To find the net change, we subtract the withdrawals from the deposit: $700 - $150 - $400. The result of this calculation is $150. This means Barry's account balance increases by $150 each month.
3. Defining the Recursive Equation
Now, let's define $f(n)$ as Barry's account balance at the end of month $n$. We know that at the end of the first month, his balance is $1900. So, $f(1) = 1900$.
4. Formulating the Recursive Equation
Since Barry's account increases by $150 each month, we can write the recursive equation as $f(n) = f(n-1) + 150$, where $n ">=" 2$. This equation states that the balance at the end of month $n$ is equal to the balance at the end of the previous month ($n-1$) plus the $150 increase.
5. Identifying the Correct Option
Therefore, the recursive equation that models Barry's account balance at the end of month $n$ is:
$f(1) = 1900$
$f(n) = f(n-1) + 150$, for $n ">=" 2$
This corresponds to option D.
### Examples
Recursive equations are useful in many real-life financial situations. For example, they can model loan balances, investment growth, or savings account balances. Understanding how to create and use recursive equations can help you predict future financial outcomes and make informed decisions about your money.
- Define $f(n)$ as the account balance at the end of month $n$, with $f(1) = $1900.
- Express the recursive equation as $f(n) = f(n-1) + $150 for $n \geq 2$, reflecting the $150 increase each month.
- The correct recursive equation is $\boxed{f(1) = 1900, f(n) = f(n-1) + 150}$ for $n \geq 2$.
### Explanation
1. Analyzing Monthly Transactions
Let's analyze Barry's monthly transactions to determine how his account balance changes each month. He deposits $700 and withdraws $150 and $400. We need to find the net change in his account balance each month.
2. Calculating Net Change
To find the net change, we subtract the withdrawals from the deposit: $700 - $150 - $400. The result of this calculation is $150. This means Barry's account balance increases by $150 each month.
3. Defining the Recursive Equation
Now, let's define $f(n)$ as Barry's account balance at the end of month $n$. We know that at the end of the first month, his balance is $1900. So, $f(1) = 1900$.
4. Formulating the Recursive Equation
Since Barry's account increases by $150 each month, we can write the recursive equation as $f(n) = f(n-1) + 150$, where $n ">=" 2$. This equation states that the balance at the end of month $n$ is equal to the balance at the end of the previous month ($n-1$) plus the $150 increase.
5. Identifying the Correct Option
Therefore, the recursive equation that models Barry's account balance at the end of month $n$ is:
$f(1) = 1900$
$f(n) = f(n-1) + 150$, for $n ">=" 2$
This corresponds to option D.
### Examples
Recursive equations are useful in many real-life financial situations. For example, they can model loan balances, investment growth, or savings account balances. Understanding how to create and use recursive equations can help you predict future financial outcomes and make informed decisions about your money.
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