To tackle this problem, let’s break down each part step-by-step.

Problem Information

• The population of a spider species at the start of year $n$ is denoted by $P_n$.
• The population at the start of the following year is given by the recursive formula: $P_{n+1} = k \cdot P_n$ where $k$ is a positive constant.
• We know:
  • $P_1 = 5$ million (start of year 1)
  • $P_2 = 6$ million (start of year 2)

Part (a): Calculate the Population at the Start of Year 3

1. Find the Value of $k$: From the recursive formula: $P_2 = k \cdot P_1$ Substituting the known values: $6 = k \cdot 5$ Solving for $k$: $k = \frac{6}{5} = 1.2$

2. Calculate $P_3$: Now that we know $k = 1.2$, we can find $P_3$ using the formula: [ P_3 = k \cdot P_2 = 1.2 \cdot 6 = 7.