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?
The initial number of rabbits is __. The common ratio is __. The number of rabbits after eight months is___.

Answer 1

To determine the rabbit population after eight months, we can use the information provided:

1. **Initial number of rabbits**: Since the rabbit population doubles every month and is 32 at the end of the first month, we can infer that the population at the start of the first month was half of 32. Therefore:
\[
\text{Initial number of rabbits} = \frac{32}{2} = 16
\]

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

3. **Number of rabbits after eight months**: The population of rabbits can be calculated using the formula for exponential growth:
\[
P(n) = P_0 \times r^n
\]
where:
- \( P(n) \) is the population after \( n \) months,
- \( P_0 \) is the initial population,
- \( r \) is the common ratio,
- \( n \) is the number of months.

Plugging in our values:
\[
P(8) = 16 \times 2^8
\]

Now calculating \( 2^8 \):
\[
2^8 = 256
\]

Therefore:
\[
P(8) = 16 \times 256 = 4096
\]

So, to summarize:
- The initial number of rabbits is **16**.
- The common ratio is **2**.
- The number of rabbits after eight months is **4096**.