An environmental organization started a new online discussion group. The group had 25 members on its first day, and added 20 new members every day after that. Which recursive sequence correctly shows the number of members the environmental organization has after n days?

Question

A) a1 = 25 an = an - 1 + 20
B) a1 = 25 an = an + 1 + 20
C) a0 = 25 an = an + 1 + 20
D) a0 = 25 an = an + 20

Answer 1

To define the recursive sequence for the number of members in the environmental organization after \( n \) days, let's analyze the options given.

• On the first day (\( n=1 \)), the number of members is 25.
• Every day after that, the organization gains 20 new members.

So, we can represent the number of members after \( n \) days with the following logic:

• For the first term, \( a_1 = 25 \).
• For each subsequent day, the total number of members increases by 20 compared to the previous day, which gives \( a_n = a_{n-1} + 20 \) for \( n \geq 2 \).

Now, let’s analyze the provided options:

A) \( a_1 = 25, ; a_n = a_{n-1} + 20 \)

This option correctly states that on the first day there are 25 members, and every day after that, it adds 20 members to the previous day's total.

B) \( a_1 = 25, ; a_n = a_n + 1 + 20 \)

This option doesn’t make sense because it references \( a_n \) in a circular manner without defining it correctly.

C) \( a_0 = 25, ; a_n = a_n + 1 + 20 \)

This is also incorrect for the same reason as B—it creates a circular reference and uses \( a_0 \) instead of \( a_1 \).

D) \( a_0 = 25, ; a_n = a_n + 20 \)

This option is incorrect as it also has a circular reference and uses \( a_0 \) incorrectly.

Given this analysis, the correct recursive sequence is:

A) \( a_1 = 25, ; a_n = a_{n-1} + 20 \)