Question

1. a new gym opens up in town. after the first month of being open, the gym has 18 members. each subsequent month, the gym gains 4 new members. model the situation above with a recursive sequence. let (a_{n}) be the total number people that have signed up for memberships (n) months after the gyms opening. 10. one hour after opening, there are 76 people at the museum. each hour a net of 36 people enter the museum. model the situation above with an explicit formula for the sequence. let (a_{n}) be the number of people at the museum (n) hours after it opens. 11. a delivery is offering a special promotion for frequent customers. on the first delivery of the month, they apply a discount of $3 to the delivery. for each subsequent delivery, the discount increases by $3. so the second delivery costs $6, the third delivery costs $9, and so on. a customer plans to use the service for a total of 11 deliveries in one month. the company wants to calculate the total money saved with discounts the customer will benefit from over the 11 deliveries. 12. the amount of medicine in a patients body is being monitored. one hour after the medicine is administered, there is a reported 4315 mg of medicine inside the patient. each hour afterward, the amount of medicine decreases by 17%. model the situation above with a recursive sequence, where (a_{n}) is the amount of medicine in the patients body (n) hours after it has been administered. 13. find the common ratio of the geometric sequence. -6, 54, -486, ... 14. find the common ratio of the geometric sequence. 7, (\frac{28}{5}), (\frac{112}{25}), ... 15. find the common difference of the arithmetic sequence. -1, -(\frac{11}{6}), -(\frac{8}{3}), ... 16. find the common difference of the arithmetic sequence. -1, -(\frac{3}{5}), -(\frac{1}{5}), ...

Explanation:

9.

Step1: Initial - condition

The gym has 18 members after the first month, so \(a_1 = 18\).

Step2: Recursive - rule

Each subsequent month it gains 4 new members. So \(a_n=a_{n - 1}+4\) for \(n\geq2\).

Step1: Initial - condition

One hour after opening, there are 76 people, so \(a_1 = 76\).

Step2: Explicit - formula

Each hour 36 new people enter. The explicit formula for an arithmetic - sequence is \(a_n=a_1+(n - 1)d\), where \(a_1\) is the first term and \(d\) is the common difference. Here \(a_1 = 76\) and \(d = 36\), so \(a_n=76+(n - 1)\times36=76 + 36n-36=36n + 40\).

Step1: Initial - discount

On the first delivery, the discount is \(d_1 = 3\).

Step2: Recursive - rule

The discount increases by 3 for each subsequent delivery. So \(d_n=d_{n - 1}+3\) for \(n\geq2\).

Step3: Total - discount

The sum of an arithmetic series \(S_n=\frac{n(d_1 + d_n)}{2}\). First, we need to find \(d_{11}\). Using the formula \(d_n=d_1+(n - 1)d\) (where \(d = 3\)), we have \(d_{11}=3+(11 - 1)\times3=3+30 = 33\). Then \(S_{11}=\frac{11\times(3 + 33)}{2}=\frac{11\times36}{2}=198\).

Answer:

\(a_1 = 18\), \(a_n=a_{n - 1}+4\) for \(n\geq2\)

10.