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

Answer 1

To model the number of members in the environmental organization as a recursive sequence, we start by defining the initial value and the growth pattern.

• On the first day (n = 1), there are 25 members. So we can write: \( a_1 = 25 \)

• Every day after that, the organization adds 20 new members. This means that the number of members after n days can be expressed in terms of the previous day's members as: \( a_n = a_{n-1} + 20 \)

Now let's evaluate the options given:

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

• This correctly expresses the initial condition and the recursive formula.

B. \( a_1 = 25 \) and \( a_n = a_n + 20 \)

• This does not make sense because it references \( a_n \) on the right side.

C. \( a_0 = 25 \) and \( a_n = a_n + 20 \)

• Again, this does not make sense because it references \( a_n \) incorrectly.

D. \( a_0 = 25 \) and \( a_n = a_n + 20 \)

• Similarly, this doesn't work as it refers back to \( a_n \) in an incorrect manner.

The correct answer is: A. \( a_1 = 25 \) and \( a_n = a_{n-1} + 20 \)