Step 1: The function \( f(x) = \frac{x}{x+1} \) is given, and we are asked to find a general pattern for \( f_n(x) \), where \( f_n(x) = f(f_{n-1}(x)) \) for \( n \geq 2 \).
Step 2: First, calculate the first few terms:
\[ f_2(x) = f(f_1(x)) = f\left(\frac{x}{x+1}\right) = \frac{\frac{x}{x+1}}{\frac{x}{x+1}+1} = \frac{x}{2x+1} \]
\[ f_3(x) = f(f_2(x)) = f\left(\frac{x}{2x+1}\right) = \frac{\frac{x}{2x+1}}{\frac{x}{2x+1}+1} = \frac{x}{3x+1} \]
Step 3: After observing the pattern for the first few terms, it becomes clear that:
\[ f_n(x) = \frac{x}{(2n-1)x+1} \]
Step 4: Now, substitute \( x = -2 \) into the formula for each \( n \):
\[ f_n(-2) = \frac{-2}{(2n-1)(-2)+1} \]
Step 5: From the general pattern \( f_n(-2) = \frac{2}{(2n-1)} \), it is clear that the product follows the form:
\[ \frac{2n}{3 \cdot 1 \cdot 5 \cdots (2n-1)}. \]
Step 1: The function \( f(x) = \frac{x}{x+1} \) is given, and we are asked to find a general pattern for \( f_n(x) \), where \( f_n(x) = f(f_{n-1}(x)) \) for \( n \geq 2 \).
Step 2: First, calculate the first few terms:
\[ f_2(x) = f(f_1(x)) = f\left(\frac{x}{x+1}\right) = \frac{\frac{x}{x+1}}{\frac{x}{x+1}+1} = \frac{x}{2x+1} \]
\[ f_3(x) = f(f_2(x)) = f\left(\frac{x}{2x+1}\right) = \frac{\frac{x}{2x+1}}{\frac{x}{2x+1}+1} = \frac{x}{3x+1} \]
Step 3: After observing the pattern for the first few terms, it becomes clear that:
\[ f_n(x) = \frac{x}{(2n-1)x+1} \]
Step 4: Now, substitute \( x = -2 \) into the formula for each \( n \):
\[ f_n(-2) = \frac{-2}{(2n-1)(-2)+1} \]
Step 5: From the general pattern \( f_n(-2) = \frac{2}{(2n-1)} \), it is clear that the product follows the form:
\[ \frac{2n}{3 \cdot 1 \cdot 5 \cdots (2n-1)}. \]
A school is organizing a debate competition with participants as speakers and judges. $ S = \{S_1, S_2, S_3, S_4\} $ where $ S = \{S_1, S_2, S_3, S_4\} $ represents the set of speakers. The judges are represented by the set: $ J = \{J_1, J_2, J_3\} $ where $ J = \{J_1, J_2, J_3\} $ represents the set of judges. Each speaker can be assigned only one judge. Let $ R $ be a relation from set $ S $ to $ J $ defined as: $ R = \{(x, y) : \text{speaker } x \text{ is judged by judge } y, x \in S, y \in J\} $.