[tex]\displaystyle \\
1+3+9 + \cdots + x =121 \\
\text{Scriem ecuatia sub forma: } \\
3^0 + 3^1 + 3^2 + \cdots + 3^n = 121~~~~~~~unde:~~3^n = x=\text{ultimul ~termen} \\ \\
\frac{3^{n+1}-1}{3-1}=121 \\ \\
\frac{3^{n+1}-1}{2}=121 \\ \\
3^{n+1}-1 = 121 \times 2 \\ \\
3^{n+1}-1 = 242 \\ \\
3^{n+1} = 242 +1 \\ \\
3^{n+1} = 243 \\ \\
3^{n+1} = 3^5 \\ \\
n+1 = 5 \\ \\
n = 5-1 = \boxed{4} \\ \\
x = 3^4 = \boxed{81} \\ \\
\texttt{Verificare: } \\
1+3+9+27+81 = 121 ~~OK![/tex]
