\( P(n) 68 \) b. Describe how the number of people who can sit at the tables changes with each step. c. Explain why \( P(3.2) \) does not make sense in this scenario. d. Define \( P \) recursively and for the \( n^{\text {th }} \) term. 4. Here is the recursive definition of a sequence: \( f(1)=35, f(n)=f(n-1)-8 \) for

Let's break down your questions step by step. ### Part b: Describe how the number of people who can sit at the tables changes with each step. Assuming \( P(n) \) represents the number of people who can sit at the tables, we can infer that as \( n \) increases (where \( n \) could represent the number of tables or some other relevant variable), the number of people who can sit at the tables also increases. The specific change in the number of people would depend on the seating capacity of each table and how that capacity scales with \( n \). For example, if each table can seat a fixed number of people, say \( k \), then the relationship could be expressed as: \[ P(n) = k \cdot n \] In this case, as \( n \) increases by 1, \( P(n) \) increases by \( k \). If the seating capacity changes in a different manner (e.g., if each additional table can seat more people than the previous one), the relationship would need to be defined accordingly. ### Part c: Explain why \( P(3.2) \) does not make sense in this scenario. In this context, \( P(n) \) likely represents a count of people, which must be a whole number. Since \( n \) typically represents a discrete quantity (like the number of tables), it should also be an integer. The value \( 3.2 \) is not an integer, and thus it does not make sense to evaluate \( P(3.2) \) because you cannot have a fraction of a table or a fraction of a person. Therefore, \( P(3.2) \) is not defined in this scenario. ### Part d: Define \( P \) recursively and for the \( n^{\text{th}} \) term. To define \( P(n) \) recursively, we need a base case and a rule for how to compute subsequent values based on previous ones. Assuming \( P(1) \) is the number of people that can sit at one table, we can define it as follows: - Base case: \( P(1) = k \) (where \( k \) is the number of people that can sit at one table) - Recursive case: \( P(n) = P(n-1) + k \) This means that to find the number of people who can sit at \( n \) tables, you take the number of people who can sit at \( n-1 \) tables and add the capacity of one more table. For the \( n^{\text{th}} \) term, if we assume \( k \) is constant, we can express it as: \[ P(n) = k \cdot n \] This gives a direct formula for the \( n^{\text{th}} \) term based on the number of tables. ### Recursive Definition of the Sequence \( f(n) \) For the sequence defined as \( f(1) = 35 \) and \( f(n) = f(n-1) - 8 \): - The base case is \( f(1) = 35 \). - The recursive case indicates that each subsequent term is 8 less than the previous term. To find the \( n^{\text{th}} \) term explicitly, we can derive it as follows: \[ f(n) = f(1) - 8(n-1) = 35 - 8(n-1) \] Simplifying this gives: \[ f(n) = 35 - 8n + 8 = 43 - 8n \] Thus, the explicit formula for the \( n^{\text{th}} \) term of the sequence is: \[ f(n) = 43 - 8n \]