[tex]\displaystyle \\
3+3^2+3^3+\cdots + 3^x = 120 \\ \\
\frac{3^{x+1}-1}{3-1}-1 = 120 \\ \\
\frac{3^{x+1}-1}{2} = 120 +1\\ \\
\frac{3^{x+1}-1}{2} = 121\\ \\
3^{x+1}-1 = 121 \cdot 2 \\ \\
3^{x+1}-1 = 242 \\ \\
3^{x+1} = 242 +1 \\ \\
3^{x+1} = 243 \\ \\
3^{x+1} = 3^5 \\ \\
x+1 = 5 \\ \\
x = 5-1 = \boxed{4} \\ \\
\texttt{Verificare:} \\ \\
3+3^2+3^3+ 3^4 = 3+9+27+81=12+27+81=39+81=120~ok! [/tex]
