b. \( 3^{n}\left(\frac{n-1}{6}\right)-3^{n}\left(\frac{n+1}{6}\right) \)

Let's simplify the expression step by step. First, we can factor out \(3^n\): \[ 3^n \left( \frac{n-1}{6} - \frac{n+1}{6} \right) \] Now focusing on the parentheses: \[ \frac{n-1}{6} - \frac{n+1}{6} = \frac{(n-1) - (n+1)}{6} = \frac{n - 1 - n - 1}{6} = \frac{-2}{6} = \frac{-1}{3} \] Now substituting this back into our expression: \[ 3^n \left( \frac{-1}{3} \right) = -\frac{3^n}{3} = -3^{n-1} \] Therefore, the final simplified form of the given expression is: \[ -\frac{3^{n}}{3} = -3^{n-1} \]