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 find the initial number of rabbits, common ratio, and the number of rabbits after eight months given the population doubles every month, we can break it down as follows:

1. **Initial Number of Rabbits:**
At the end of the first month, the population is 32 rabbits. Since the population doubles every month, we can find the initial number (at the beginning of the first month) by halving the month-end population:
\[
\text{Initial number} = \frac{32}{2} = 16
\]

2. **Common Ratio:**
The common ratio for a population that doubles is 2.

3. **Number of Rabbits After Eight Months:**
If the population doubles every month, we can find the number of rabbits after \( n \) months using the formula:
\[
\text{Population after } n \text{ months} = \text{Initial population} \times (\text{common ratio})^n
\]
In this case, the initial population is 16, the common ratio is 2, and \( n \) is 8 months. Therefore:
\[
\text{Population after 8 months} = 16 \times 2^8
\]
Calculating \( 2^8 \):
\[
2^8 = 256
\]
Now, substituting back into the population formula:
\[
\text{Population after 8 months} = 16 \times 256 = 4096
\]

Thus, we have:
- The initial number of rabbits is **16**.
- The common ratio is **2**.
- The number of rabbits after eight months is **4096**.