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 solve this problem, we'll start with the given information:

1. The rabbit population doubles every month.
2. At the end of the first month, there are 32 rabbits.

### Step 1: Determine the initial population
Since the population doubles every month, we can determine the initial population (at the start of the first month) by halving the population at the end of the first month:
- Initial population = \( \frac{32 \text{ rabbits}}{2} = 16 \text{ rabbits} \)

### Step 2: Identify the common ratio
The common ratio in this case (since the population doubles each month) is:
- Common ratio = \( 2 \)

### Step 3: Calculate the number of rabbits after eight months
Since the population doubles every month, we can calculate the population at the end of each month using the formula for exponential growth:
- Population at month \( n \) = Initial population \( \times \) (common ratio)^{(n - 1)}

For 8 months:
- Population at end of 8 months = Initial population \( \times 2^{(8 - 1)} \)
- Population at end of 8 months = \( 16 \times 2^7 \)

Calculating \( 2^7 \):
- \( 2^7 = 128 \)

Thus,
- Population at end of 8 months = \( 16 \times 128 = 2048 \)

### Final Values
- The initial number of rabbits is **16**.
- The common ratio is **2**.
- The number of rabbits after eight months is **2048**.

So, to summarize:
- The initial number of rabbits is 16
- The common ratio is 2
- The number of rabbits after eight months is 2048