Question

A new gym opens up in town. After the first month of being open, the gym has 11
members. Each subsequent month, the gym gains 2
new members.

Model the situation above with a recursive sequence. Let
an be the total number people that have signed up for memberships
months after the gym's openin

Answer 1

To model the situation of the gym's membership growth with a recursive sequence, we can define the sequence \( a_n \) where \( n \) represents the number of months after the gym's opening.

### Base Case:
- Initially, after 1 month, the gym has 11 members. So, we set:
\[
a_1 = 11
\]

### Recursive Formula:
- Each subsequent month, the gym gains 2 new members. Therefore, the total number of members in the \( n \)-th month can be expressed in terms of the previous month's total:
\[
a_n = a_{n-1} + 2 \quad \text{for } n \geq 2
\]

### Summary of the Recursive Sequence:
1. \( a_1 = 11 \) (Base Case)
2. \( a_n = a_{n-1} + 2 \) for \( n \geq 2 \)

### Example Calculation:
To see how this works, let's calculate the total number of members for a few months:
- For \( n = 2 \):
\[
a_2 = a_1 + 2 = 11 + 2 = 13
\]
- For \( n = 3 \):
\[
a_3 = a_2 + 2 = 13 + 2 = 15
\]
- For \( n = 4 \):
\[
a_4 = a_3 + 2 = 15 + 2 = 17
\]

With this recursive formula, you can determine the total number of gym members for any month after the first month.