Question

What is the general term equation, an for the arithmetic sequence 13,9,5,1 and what is the 21 term

Answer 1

$\bf 13~~,~~\stackrel{13-4}{9}~~,~~\stackrel{9-4}{5}~~,~~\stackrel{5-4}{1}~~,~~...$

so as we can see, the "common difference" is -4, namely we're "adding" -4 to get the next term, and the first term is 13.

$\bf n^{th}\textit{ term of an arithmetic sequence} \\\\ a_n=a_1+(n-1)d\qquad \begin{cases} n=n^{th}\ term\\ a_1=\textit{first term's value}\\ d=\textit{common difference}\\ ----------\\ d=-4\\ a_1=13\\ n=21 \end{cases} \\\\\\ \stackrel{general~form}{a_n=13+(n-1)(-4)} \\\\\\ a_{21}=13+(21-1)(-4)\implies a_{21}=13-80\implies a_{21}=-67$