Question

Geometric Sequences Practice
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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
.
Remaining Attempts : 3

Answer 1

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:

1. The initial number of rabbits at the end of the first month \( a_1 = 32 \).
2. The common ratio \( r = 2 \) (since the population doubles each month).
3. 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:

1. Calculate \( 2^{7} = 128 \).
2. 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 \( 4096 \).