Question
b. \( 3^{n}\left(\frac{n-1}{6}\right)-3^{n}\left(\frac{n+1}{6}\right) \)
Ask by Clarke Weber. in Canada
Feb 21,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The simplified expression is \( -3^{n-1} \).
Solution
Calculate or simplify the expression \( 3^{n}\left(\frac{n-1}{6}\right)-3^{n}\left(\frac{n+1}{6}\right) \).
Simplify the expression by following steps:
- step0: Solution:
\(3^{n}\left(\frac{n-1}{6}\right)-3^{n}\left(\frac{n+1}{6}\right)\)
- step1: Remove the parentheses:
\(3^{n}\times \frac{n-1}{6}-3^{n}\left(\frac{n+1}{6}\right)\)
- step2: Remove the parentheses:
\(3^{n}\times \frac{n-1}{6}-3^{n}\times \frac{n+1}{6}\)
- step3: Multiply the terms:
\(\frac{3^{n}\left(n-1\right)}{6}-3^{n}\times \frac{n+1}{6}\)
- step4: Multiply the terms:
\(\frac{3^{n}\left(n-1\right)}{6}-\frac{3^{n}\left(n+1\right)}{6}\)
- step5: Transform the expression:
\(\frac{3^{n}\left(n-1\right)-3^{n}\left(n+1\right)}{6}\)
- step6: Multiply the terms:
\(\frac{n\times 3^{n}-3^{n}-3^{n}\left(n+1\right)}{6}\)
- step7: Multiply the terms:
\(\frac{n\times 3^{n}-3^{n}-\left(n\times 3^{n}+3^{n}\right)}{6}\)
- step8: Subtract the terms:
\(\frac{-2\times 3^{n}}{6}\)
- step9: Rewrite the fraction:
\(-\frac{2\times 3^{n}}{6}\)
- step10: Reduce the fraction:
\(-3^{n-1}\)
The simplified expression is \( -3^{n-1} \).
Answered by UpStudy AI and reviewed by a Professional Tutor
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Simplify this solution Beyond the Answer
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: \[ -\frac{3^{n}}{3} = -3^{n-1} \]
