Question

2+2*3+2*3^2+2*3^3+...+2*3^2011=x*81^502-1

Answer 1

[tex]2+2\cdot 3+2\cdot3^2+...+2\cdot 3^{2011}=x\cdot81^{502}-1\\ 2(1+3+3^2+...+3^{2011})=x\cdot 81^{502}-1\\ Rezolvam\ separat\ suma:\\ S=1+3+3^2+3^3+...+3^{2011}|\cdot 3\\ 3S=3+3^2+3^3+...+3^{2012}\\ .................................................-\\ 3S-S=3^{2012}-1\\ 2S=3^{2012}-1\\ S=\frac{3^{2012}-1}{2}\\ Revenind: 2\cdot \frac{3^{2012}-1}{2}=x\cdot 81^{502}-1\\ \\ 3^{2012}-1=x\cdot81^{502}-1\\ x\cdot (3^4)^{502}=3^{2012}\\ x\cdot 3^{2008}=3^{2012}\\ x=3^4\Rightarrow \boxed{x=81}[/tex]