\begin{tabular}{l}\( \begin{array}{l}a+b+c \\ \frac{a+b+c}{b+c}=-1\end{array} \) \\ Determine the \( 50^{\text {th }} \) term of the above pattern: \\ \hline\end{tabular}

To solve the problem, we need to analyze the given equations and find a pattern that allows us to determine the \( 50^{\text{th}} \) term. 1. **Extract Known Conditions:** - We have two equations: 1. \( a + b + c \) 2. \( \frac{a + b + c}{b + c} = -1 \) 2. **Solve the Second Equation:** - From the second equation, we can rewrite it as: \[ a + b + c = - (b + c) \] - This simplifies to: \[ a + b + c = -b - c \] - Rearranging gives: \[ a + 2b + 2c = 0 \] - Thus, we can express \( a \) in terms of \( b \) and \( c \): \[ a = -2b - 2c \] 3. **Substituting Back:** - Now we can substitute \( a \) back into the first equation: \[ (-2b - 2c) + b + c = -b - c \] - Simplifying this gives: \[ -b - c = -b - c \] - This is always true, indicating that \( a, b, c \) can take on various values as long as they satisfy the relationship derived. 4. **Finding the Pattern:** - The equations suggest a linear relationship among \( a, b, c \). If we let \( b = n \) and \( c = m \), we can express \( a \) as: \[ a = -2n - 2m \] - The terms \( a, b, c \) can be represented in a sequence based on \( n \) and \( m \). 5. **Determining the \( 50^{\text{th}} \) Term:** - If we assume a simple pattern where \( a, b, c \) are integers, we can generate terms based on \( n \) and \( m \). - For example, if we let \( n \) and \( m \) increase sequentially, we can find the \( 50^{\text{th}} \) term by substituting \( n = 50 \) and \( m = 50 \) into our expressions. 6. **Calculating the \( 50^{\text{th}} \) Term:** - Let’s calculate: \[ a = -2(50) - 2(50) = -100 - 100 = -200 \] - Thus, the \( 50^{\text{th}} \) term can be represented as: \[ a + b + c = -200 + 50 + 50 = -100 \] Therefore, the \( 50^{\text{th}} \) term of the pattern is \(-100\).