Question

A rabbit population doubles every month. If the population of rabbits is 32 at the end of the first month, how many rabbits will there be after eight months?(2 points)
The initial number of rabbits is
. The common ratio is
. The number of rabbits after eight months is

Answer 1

To solve the problem, let's break it down step-by-step.

1. **Initial Population**: The population at the end of the first month is given as 32 rabbits. Since the population doubles every month, the initial population at the start (before any doubling) would be half of this:
\[
\text{Initial population} = \frac{32}{2} = 16
\]

2. **Common Ratio**: The common ratio is the factor by which the population increases each month. Since the population doubles, the common ratio is:
\[
\text{Common ratio} = 2
\]

3. **Population After Eight Months**: The formula for calculating the population after a certain number of months, given the initial population and the common ratio, is:
\[
P = P_0 \times r^n
\]
where:
- \(P_0\) is the initial population,
- \(r\) is the common ratio,
- \(n\) is the number of months.

After eight months, we want to find \(P\):
\[
P = 16 \times 2^8
\]
We need to calculate \(2^8\):
\[
2^8 = 256
\]
Now, substituting back:
\[
P = 16 \times 256 = 4096
\]

Therefore, the answers are:

- The initial number of rabbits is **16**.
- The common ratio is **2**.
- The number of rabbits after eight months is **4096**.