\(x+3x+3^2x+3^3x+...+3^{49}x=3^{100}-3^{50}\)
\(\Leftrightarrow x\left(1+3+3^2+3^3+...+3^{49}\right)=3^{100}-3^{50}\)
Đặt \(A=1+3+3^2+3^3+...+3^{49}\)
\(\Leftrightarrow3A=3+3^2+3^3+3^4+3^{50}\)
\(\Leftrightarrow3A-A=2A=3^{50}-1\)
\(\Leftrightarrow A=\frac{3^{50}-1}{2}\)
\(\Rightarrow\frac{x\left(3^{50}-1\right)}{2}=3^{100}-3^{50}\)
\(\Leftrightarrow x\left(3^{50}-1\right)=2.3^{100}-2.3^{50}\)
\(\Leftrightarrow x=\frac{2.3^{100}-2.3^{50}}{3^{50}-1}\)
\(x+3x+3^2x+3^3x+...+3^{49}x=3^{100}-3^{50}\)
\(\Leftrightarrow x\left(1+3+3^2+...+3^{49}\right)=3^{100}-3^{50}\)
\(\Leftrightarrow x\left(\frac{3^{50}-1}{2}\right)=3^{100}-3^{50}\)
\(\Leftrightarrow x\left(3^{50}-1\right)=3^{102}-3^{52}\)
\(\Rightarrow x=\frac{3^{102}-3^{52}}{3^{50}-1}=\frac{3^{52}\left(3^{50}-1\right)}{3^{50}-1}=3^{52}\)
