a)
\(3^{n+1}+5.3^{n-2}=2592\)
\(\Rightarrow3^{n+1}+5.3^{n+1-3}=2592\)
\(\Rightarrow3^{n+1}+\dfrac{1}{27}.5.3^{n+1}=2592\)
\(\Rightarrow3^{n+1}+\dfrac{5}{27}.3^{n+1}=2592\)
\(\Rightarrow3^{n+1}.\left(\dfrac{5}{27}+1\right)=2592\)
\(\Rightarrow3^{n+1}.\dfrac{32}{27}=2592\)
\(\Rightarrow3^{n+1}=2187\)
\(\Rightarrow3^{n+1}=3^7\)
\(\Rightarrow n+1=7\)
\(\Rightarrow n=6\)
b)
\(3^{n+2}.5.3^{n-1}=864\)
\(\Rightarrow3^{n+2}+\dfrac{1}{27}.5.3^{n+2}=864\)
\(\Rightarrow3^{n+2}\left(\dfrac{5}{27}+1\right)=864\)
\(\Rightarrow3^{n+2}.\dfrac{32}{27}=864\)
\(\Rightarrow3^{n+2}=729\)
\(\Rightarrow3^{n+2}=3^6\)
\(\Rightarrow n+2=6\)
\(\Rightarrow n=4\)