Nth term

whats the nth term of 2,8,18,32

$\small{ \begin{array}{lrrrrrrrrrr} & {\color{red}d_0 = 2} && 8 && 18 && 32 && 50 && \cdots \\ \text{1. Difference } && {\color{red}d_1 = 6} && 10 && 14 && 18 && \cdots \\ \text{2. Difference } &&& {\color{red}d_2 = 4} && 4 && 4 && \cdots \\ \end{array} }$

$\boxed{~ \begin{array}{rcl} a_n &=& \dbinom{n-1}{0}\cdot {\color{red}d_0 } + \dbinom{n-1}{1}\cdot {\color{red}d_1 } + \dbinom{n-1}{2}\cdot {\color{red}d_2 } \end{array} ~}$

$\begin{array}{|rcll|} \hline a_n &=& \dbinom{n-1}{0}\cdot {\color{red} 2 } + \dbinom{n-1}{1}\cdot {\color{red} 6 } + \dbinom{n-1}{2}\cdot {\color{red} 4 } \\\\ && \binom{n-1}{0} = 1 \\ && \binom{n-1}{1} = n-1 \\ && \binom{n-1}{2} = \left( \frac{n-1}{2} \right) \cdot \left( \frac{n-2}{1} \right)\\\\ a_n &=& {\color{red} 2 } + (n-1)\cdot {\color{red} 6 } + \left( \frac{n-1}{2} \right) \cdot (n-2) \cdot {\color{red} 4 } \\ &=& {\color{red} 2 } + (n-1)\cdot {\color{red} 6 } + (n-1) \cdot (n-2) \cdot {\color{red} 2 } \\ &=& 2 + 6n-6 + (2n-2) \cdot (n-2) \\ &=& -4 + 6n + 2n^2-4n-2n+4 \\ \mathbf{a_n} & \mathbf{=} & \mathbf{2n^2} \\ \hline \end{array}$

Example:

$\begin{array}{|rclcl|} \hline a_1 &=& 2\cdot 1^2 &=& 2 \\ a_2 &=& 2\cdot 2^2 &=& 8 \\ a_3 &=& 2\cdot 3^2 &=& 18 \\ a_4 &=& 2\cdot 4^2 &=& 32 \\ a_5 &=& 2\cdot 5^2 &=& 50 \\ \dots \\ \mathbf{a_n} &\mathbf{=}& \mathbf{2\cdot n^2} \\ \hline \end{array}$