To understand how Jamie reached her solution of having 3,200 clients after 6 months, we can break down the expression given for her number of clients: \( 50 \cdot 2^m \), where \( m \) represents the number of months.
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Initial Number of Clients: Jamie starts with 50 clients.
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Exponential Growth: Each month, her number of clients doubles. This is represented by the multiplication by \( 2 \) raised to the power of \( m \), where \( m \) is the number of months that have passed.
The formula \( 50 \cdot 2^m \) means that after \( m \) months, her clients can be calculated as follows:
- After 0 months (\( m = 0 \)): \[ 50 \cdot 2^0 = 50 \cdot 1 = 50 \ \text{clients} \]
- After 1 month (\( m = 1 \)): \[ 50 \cdot 2^1 = 50 \cdot 2 = 100 \ \text{clients} \]
- After 2 months (\( m = 2 \)): \[ 50 \cdot 2^2 = 50 \cdot 4 = 200 \ \text{clients} \]
- After 3 months (\( m = 3 \)): \[ 50 \cdot 2^3 = 50 \cdot 8 = 400 \ \text{clients} \]
- After 4 months (\( m = 4 \)): \[ 50 \cdot 2^4 = 50 \cdot 16 = 800 \ \text{clients} \]
- After 5 months (\( m = 5 \)): \[ 50 \cdot 2^5 = 50 \cdot 32 = 1600 \ \text{clients} \]
- After 6 months (\( m = 6 \)): \[ 50 \cdot 2^6 = 50 \cdot 64 = 3200 \ \text{clients} \]
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Calculating After 6 Months: To find the number of clients after 6 months, we substitute \( m = 6 \) into the expression: \[ 50 \cdot 2^6 = 50 \cdot 64 = 3200 \]
Thus, after 6 months, Jamie reaches 3,200 clients by using the mathematical model \( 50 \cdot 2^m \) correctly where she calculates the number of clients based on exponential growth, leading to the conclusion that she could have reached that figure after doubling her clients each month.
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