Answer :
- Calculate the net change in Barry's account each month: $700 - $150 - $400 = $150.
- Determine the initial value: $f(1) = 1900$.
- Write the recursive equation: $f(n) = f(n-1) + 150$, for $n \geq 2$.
- The correct recursive equation is $\boxed{B}$.
### Explanation
1. Analyzing the Problem
Let's analyze Barry's monthly transactions to determine the recursive equation that models his account balance.
2. Calculating Net Monthly Change
Barry deposits $700 and withdraws $150 and $400 each month. The net change in his account each month is the deposit minus the withdrawals: $700 - $150 - $400 = $150.
3. Determining the Initial Value
The account balance at the end of the first month is given as $1,900. This serves as the initial value for our recursive equation, so $f(1) = 1900$.
4. Formulating the Recursive Equation
The recursive equation models the account balance at the end of month $n$ as the balance at the end of the previous month ($n-1$) plus the net change in the account each month. Therefore, the recursive equation is $f(n) = f(n-1) + 150$, for $n ">=" 2$.
5. Selecting the Correct Option
Comparing our derived recursive equation with the given options, we find that option B matches our result: $f(1) = 1,900$ and $f(n) = f(n-1) + 150$, for $n ">=" 2$.
6. Final Answer
Therefore, the correct recursive equation that models Barry's account balance at the end of month $m$ is option B.
### Examples
Recursive equations are useful in many real-life situations, such as modeling population growth, calculating compound interest, or tracking inventory levels. For example, a business might use a recursive equation to predict its revenue each quarter based on the previous quarter's revenue and a growth rate. Understanding recursive equations helps in making informed decisions and predictions in various fields.
- Determine the initial value: $f(1) = 1900$.
- Write the recursive equation: $f(n) = f(n-1) + 150$, for $n \geq 2$.
- The correct recursive equation is $\boxed{B}$.
### Explanation
1. Analyzing the Problem
Let's analyze Barry's monthly transactions to determine the recursive equation that models his account balance.
2. Calculating Net Monthly Change
Barry deposits $700 and withdraws $150 and $400 each month. The net change in his account each month is the deposit minus the withdrawals: $700 - $150 - $400 = $150.
3. Determining the Initial Value
The account balance at the end of the first month is given as $1,900. This serves as the initial value for our recursive equation, so $f(1) = 1900$.
4. Formulating the Recursive Equation
The recursive equation models the account balance at the end of month $n$ as the balance at the end of the previous month ($n-1$) plus the net change in the account each month. Therefore, the recursive equation is $f(n) = f(n-1) + 150$, for $n ">=" 2$.
5. Selecting the Correct Option
Comparing our derived recursive equation with the given options, we find that option B matches our result: $f(1) = 1,900$ and $f(n) = f(n-1) + 150$, for $n ">=" 2$.
6. Final Answer
Therefore, the correct recursive equation that models Barry's account balance at the end of month $m$ is option B.
### Examples
Recursive equations are useful in many real-life situations, such as modeling population growth, calculating compound interest, or tracking inventory levels. For example, a business might use a recursive equation to predict its revenue each quarter based on the previous quarter's revenue and a growth rate. Understanding recursive equations helps in making informed decisions and predictions in various fields.
