The first term of a geometric series is 2, the nth term is 486 and the sum of the n terms is 728. Find r and n

The first term of a geometric series is 2, the nth term is 486 and the sum of the n terms is 728.

Find r and n

$$\small{
\begin{array}{lrcl}
(1) & a_1&=&\mathbf{a=2}\\\\
(2) & a_n &=& a\cdot r^{n-1} \\
& 486 &=& 2 \cdot r^{n-1} \\
& \mathbf{r^{n-1}} &\mathbf{=} & \mathbf{ 243 } \\\\
(3) & s_n - s_{n-1} &=& a_n \\
& 728 - s_{n-1} &=& 486 \\
& s_{n-1} &=& 728-486 \\
& \mathbf{s_{n-1}} & \mathbf{=} & \mathbf{242} \\\\
(4) & s_{n-1} &=& a\cdot \dfrac{ r^{n-1} -1 }{ r-1 } \\\\
& 242 &=& 2 \cdot \dfrac{ 243-1 }{ r-1 } \\\\
& 242 &=& 2 \cdot \dfrac{ 242 }{ r-1 } \\\\
& r-1 &=& 2\\
& \mathbf{r} & \mathbf{=} & \mathbf{3}\\\\
(5) & r^{n-1} & = & 243 \\\\
& 3^{n-1} &=& 3^5 \\
& n-1 &=& 5 \\
& \mathbf{n} &\mathbf{=} & \mathbf{6}
\end{array}
}$$

r = 3 and n = 6

$$\small{
\begin{array}{rclcr}
\text{geometric sequence: }
a_1 &=& 2 \cdot 3^0 &=& 2\\
a_2 &=& 2\cdot 3^1 &=& 6 \\
a_3 &=& 2\cdot 3^2 &=& 18\\
a_4 &=& 2\cdot 3^3 &=& 54\\
a_5 &=& 2\cdot 3^4 &=& 162\\
a_6 &=& 2\cdot 3^5 &=& 486\\
\hline
s_6 &&&= & 728
\end{array}
}$$ 

check:

$$\small{
a_n=a_6 = 2\cdot 3^5 = 486 \quad \text{( okay)}
}$$