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:
- \( a_1 = 11 \) (Base Case)
- \( a_n = a_{n-1} + 2 \) 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.
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