\(\frac 13 + \frac1{3^2} + \frac 1{3^3} + ... \infty\)
\(= \frac{1/3}{1 - 1/3}\)
\(= \frac 12\)
\((0.16)^{\log_{2.5}(\frac 13 + \frac 1{3^2} + \frac 1{3^3} + ...\infty)}\)
\(= [(0.4)^2]^{\log_{\frac 52}\frac 12}\)
\(= (0.4)^{\log_{\frac 52}\frac 14}\)
\(= (\frac 14) ^{\log_{\frac 52}\frac 25}\) \((\because a^{\log _bc} = c^{\log_ba})\)
\(= (4) ^{-\log_\frac 52 \frac 25}\)
\(= (4) ^{\log_\frac 25 \frac 25}\)
\(=4\)
