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 determine the correct recursive sequence, we need to analyze how the number of members grows over days.

• On Day 1 (n=1), there are 25 members.
• On Day 2 (n=2), the organization adds 20 new members, so there are 25 + 20 = 45 members.
• On Day 3 (n=3), it adds another 20 members, bringing the total to 45 + 20 = 65 members.

From observing the pattern, we can define the sequence as follows:

1. Let \( a_n \) be the number of members after \( n \) days.
2. The first term, \( a_1 \), is 25 members.
3. For each subsequent day, the number of members increases by 20 - that is, \( a_n = a_{n-1} + 20 \).

Now we can compare this definition with the options provided:

A. \( a_1 = 25 \) and \( a_n = a_{n-1} + 20 \) → This matches our observations. B. \( a_1 = 25 \) and \( a_n = a_n + 1 + 20 \) → This does not match. C. \( a_0 = 25 \) and \( a_n = a_n + 1 + 20 \) → This does not match. D. \( a_0 = 25 \) and \( a_n = a_n + 20 \) → This does not match.

The correct answer is:

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