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:

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