Liberal Arts Math Question #3

What is the general-term equation, an, for the arithmetic sequence below,

and what is the 23rd term of this sequence?
$\begin{array}{|l|r|r|r|} \hline n & 1 & 2 & 3 \\ \hline a_n & 2 & -3& -8 \\ \hline \end{array}$

$\begin{array}{|rcll|} \hline a_n &=& a_1 + (n-1)\cdot d \quad & | \quad a_1 = 1 \\ && \quad & | \quad d = a_2-a_1\\ && \quad & | \quad = -3-2 \\ && \quad & | \quad = -5 \\ a_n &=& 2 + (n-1)(-5) \\ a_n &=& 2 -5n + 5 \\ a_n &=& 7 -5n \\\\ a_{23} &=& 7 - 5\cdot 23 \\ a_{23} &=& -108 \\ \hline \end{array}$