Questions: Consider a population that demonstrates linear growth: ... P5=233, P6=278, P7=323, P8=368, ... What is the common difference for this linear pattern? d= What is the initial population? P0=

Solution Steps

To solve the given problem, we need to determine the common difference \( d \) and the initial population \( P_0 \) for the linear growth pattern.

1. Common Difference \( d \): The common difference in a linear sequence can be found by subtracting any term from the subsequent term.
2. Initial Population \( P_0 \): Using the common difference and one of the given terms, we can backtrack to find the initial population.

The common difference \( d \) in a linear sequence can be found by subtracting any term from the subsequent term. Using the given values: \[ d = P_6 - P_5 = 278 - 233 = 45 \]

Using the formula for the \( n \)-th term of an arithmetic sequence: \[ P_n = P_0 + n \cdot d \] We can rearrange to find \( P_0 \): \[ P_0 = P_n - n \cdot d \] Using \( P_5 \): \[ P_0 = P_5 - 5 \cdot d = 233 - 5 \cdot 45 = 233 - 225 = 8 \]

Final Answer