[tex]E = 3^{2n-1}+2^{n+1}[/tex]
[tex]\text{Notez: }n = m+1,\,\, m\in \mathbb{N}[/tex]
[tex]\Rightarrow E = 3^{2m+1}+2^{m+2}[/tex]
[tex]\Rightarrow E = 3\cdot 3^{2m}+4\cdot 2^{m}[/tex]
[tex]\Rightarrow E = 3\cdot 9^m+4\cdot 2^{m}[/tex]
[tex]\Rightarrow E = 3\cdot (7+2)^m+4\cdot 2^{m}[/tex]
[tex]\Rightarrow E = 3\cdot (M_7+2^{m})+4\cdot 2^{m}[/tex]
[tex]\Rightarrow E = M_7+3\cdot 2^m+4\cdot 2^m[/tex]
[tex]\Rightarrow E = M_7+2^m\cdot (3+4)[/tex]
[tex]\Rightarrow E = M_7+2^m\cdot 7[/tex]
[tex]\Rightarrow E = M_7+M_7[/tex]
[tex]\Rightarrow E = M_7,\,\,\forall m \in \mathbb{N}[/tex]
[tex]\Rightarrow \boxed{E = M_{7},\,\,\forall n \in \mathbb{N}^*}[/tex]
