Divide[n,5]*Divide[\(40)Power[\(40)Divide[1,5]\(41),Log[5,n]]-1\(41),-Divide[4,5]]
You asked:
Evaluate the expression: \(\frac{n}{5} \cdot \frac{{\left( \frac{1}{5} \right)}^{\log_{5}\left( n \right)} - 1}{-\left( \frac{4}{5} \right)}\)
MathBot Answer:
Evaluated
\(\displaystyle \frac{n}{5} \cdot \frac{{\left( \frac{1}{5} \right)}^{\log_{5}\left( n \right)} - 1}{-\left( \frac{4}{5} \right)} = - \frac{n \left(-1 + \left(\frac{1}{5}\right)^{\frac{\log{\left(n \right)}}{\log{\left(5 \right)}}}\right)}{4} \)
Expanded
\[\frac{n}{5} \cdot \frac{{\left( \frac{1}{5} \right)}^{\log_{5}\left( n \right)} - 1}{-\left( \frac{4}{5} \right)} = \frac{n}{4} - \frac{1}{4}\]
Factored
\[\frac{n}{5} \cdot \frac{{\left( \frac{1}{5} \right)}^{\log_{5}\left( n \right)} - 1}{-\left( \frac{4}{5} \right)} = \frac{5^{- \log{\left(n \right)}} n \left(n - 1\right)}{4}\]