To solve the problem of the rabbit population, we can use the formula for a geometric sequence:
\[ a_n = a_1 \cdot r^{(n-1)} \]
where:
- \( a_n \) is the amount after \( n \) months,
- \( a_1 \) is the initial amount,
- \( r \) is the common ratio, and
- \( n \) is the number of terms (months in this case).
From the information given:
- The initial number of rabbits at the end of the first month \( a_1 = 32 \).
- The common ratio \( r = 2 \) (since the population doubles each month).
- We want to find the number of rabbits after 8 months, which means \( n = 8 \).
Using the formula:
\[ a_8 = 32 \cdot 2^{(8-1)} \]
Calculating it step by step:
- Calculate \( 2^{7} = 128 \).
- Now multiply by the initial amount:
\[ a_8 = 32 \cdot 128 = 4096 \]
So the final answers are:
- The initial number of rabbits is \( 32 \).
- The common ratio is \( 2 \).
- The number of rabbits after eight months is \( 4096 \).