10. Which of the following is UNLIKE the others with respect to the rate of change (common difference)?

Question

A) 2, 8, 12, 20, 26
B) f(x) = 6x +1
C) f(x) = 6x − 2
D) an = −5 + 6(n − 1)

11. 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 where x is the number of months since March. What will the population of bunnies be after 5 months?
A) 2916
B) 243
C) 420
D) 180

12. Convert the recursive form a1 = 5, an = 2an−1 to the explicit form.
A) an = 5(7)^n
B) an = 5(2)^n-1
C) an = 5(2)^n+1
D) an = 2(5)^n-1

13. Given: an = 12(1/3)^n-1 match the term to its answer. Answers may be used more than once.
a2 - ________
Common Ratio - ________
a1 - _______
A) 1/3
B) 4
C) 12

Answer 1

Let's go through each question one by one.

Question 10:

You need to find the one that is UNLIKE the others concerning the rate of change (common difference).

A) 2, 8, 12, 20, 26:

• Differences: 6 (8-2), 4 (12-8), 8 (20-12), 6 (26-20); not constant.

B) f(x) = 6x + 1:

• This is a linear function with a constant rate of change of 6.

C) f(x) = 6x - 2:

• This is also a linear function with a constant rate of change of 6.

D) an = −5 + 6(n − 1):

• This is an arithmetic sequence with a common difference of 6.

Answer: A) because the sequence in A does not have a constant common difference.

Question 11:

Using the provided equation \( P = 12(3)^x \) to find the population after 5 months (x = 5).

\[ P = 12(3)^5 = 12(243) = 2916 \]

Answer: A) 2916.

Question 12:

Convert the recursive form \( a_1 = 5, , a_n = 2a_{n−1} \) to an explicit form.

1. The sequence starts with \( a_1 = 5 \).
2. The next term is \( a_2 = 2a_1 = 2(5) \).
3. Continuing, \( a_n = 5(2)^{n-1} \).

Thus, the explicit form is:

Answer: B) \( a_n = 5(2)^{n-1} \).

Question 13:

Given \( a_n = 12(1/3)^{n-1} \):

• a2 = \( 12(1/3)^{2-1} = 12(1/3) = 4 \)
• Common Ratio = \( 1/3 \)
• a1 = \( 12(1/3)^{1-1} = 12(1) = 12 \)

Match the terms to their answers:

• a2 - B) 4
• Common Ratio - A) 1/3
• a1 - C) 12

Final matches:

• a2 - B) 4
• Common Ratio - A) 1/3
• a1 - C) 12