The population of bunnies in March is 12. A researcher notices that the population is tripling each month. The population can be represented by the equation P = 12(3)^x

Question

where x is the number of months since March. What will the population of bunnies be after 5 months? (1 point)
Responses

420

243

180

2916

Answer 1

To find the population of bunnies after 5 months, we will use the equation provided:

\[ P = 12(3)^x \]

where \( x \) is the number of months since March. Since we want to find the population after 5 months, we will set \( x = 5 \):

\[ P = 12(3)^5 \]

First, calculate \( 3^5 \):

\[ 3^5 = 243 \]

Now, substitute this back into the equation:

\[ P = 12 \times 243 \]

Now multiply:

\[ P = 2916 \]

Thus, the population of bunnies after 5 months will be 2916.

Answer 2

Convert the recursive form a1=5, an=2an−1
to the explicit form. (1 point)
Responses

an=2(5)^n−1

an=3(7)^n

an=5(2)^n+1

an=5(2)^n−1

Answer 3

To convert the recursive sequence given by:

• \( a_1 = 5 \)
• \( a_n = 2a_{n-1} \)

to explicit form, we will look for a pattern.

1. Calculate a few terms to identify the pattern:

   • \( a_1 = 5 \)
   • \( a_2 = 2a_1 = 2 \times 5 = 10 \)
   • \( a_3 = 2a_2 = 2 \times 10 = 20 \)
   • \( a_4 = 2a_3 = 2 \times 20 = 40 \)
   • \( a_5 = 2a_4 = 2 \times 40 = 80 \)

2. Identify the pattern: From the calculations, we can see that:

   • \( a_1 = 5 = 5 \cdot 2^0 \)
   • \( a_2 = 10 = 5 \cdot 2^1 \)
   • \( a_3 = 20 = 5 \cdot 2^2 \)
   • \( a_4 = 40 = 5 \cdot 2^3 \)
   • \( a_5 = 80 = 5 \cdot 2^4 \)

   It appears that the formula has the form: \[ a_n = 5 \cdot 2^{n-1} \]

3. Explicit formula: Therefore, the explicit formula for the sequence is: \[ a_n = 5(2)^{n-1} \]

Now let's match this to the provided response options:

• \( a_n = 2(5)^{n-1} \)
• \( a_n = 3(7)^{n} \)
• \( a_n = 5(2)^{n+1} \)
• \( a_n = 5(2)^{n-1} \)

The correct response is \( a_n = 5(2)^{n-1} \).

Answer 4

Given: an=12(1/3)^n−1
match the term to its answer. Answers may be used more than once.(3 points)

a2

Common Ratio

a1

Anwers

1/3

4

12

Answer 5

To start with, we have the explicit equation for the sequence:

\[ a_n = 12 \left(\frac{1}{3}\right)^{n-1} \]

Let's identify the values needed:

1. Calculate \( a_1 \): \[ a_1 = 12 \left(\frac{1}{3}\right)^{1-1} = 12 \left(\frac{1}{3}\right)^0 = 12 \times 1 = 12 \]

2. Calculate \( a_2 \): \[ a_2 = 12 \left(\frac{1}{3}\right)^{2-1} = 12 \left(\frac{1}{3}\right)^1 = 12 \times \frac{1}{3} = 4 \]

3. Identify the common ratio: The base of the expression \(\frac{1}{3}\) indicates the common ratio. Therefore, the common ratio is:

\[ \text{Common Ratio} = \frac{1}{3} \]

Now that we've determined the values, we can match them to the provided options:

1. \( a_2 = 4 \) (from our calculation)
2. Common Ratio = \( \frac{1}{3} \) (from our identification)
3. \( a_1 = 12 \) (from our calculation)

Here's the matching:

• \( a_2 \) → 4
• Common Ratio → \( \frac{1}{3} \)
• \( a_1 \) → 12

If presented with the answers, it matches as follows:

• \( a_2 \) → 4
• Common Ratio → \( \frac{1}{3} \)
• \( a_1 \) → 12.