Question

What is the 33th term of this arithmetic sequence 12,7,2,-3,-8

Answer 1

$\bf 12~~,~~\stackrel{12-5}{7}~~,~~\stackrel{7-5}{2}~~,~~\stackrel{2-5}{-3}~~,~~\stackrel{-3-5}{-8}~~...$

so, as you can see, the "common difference" is -5, namely to get the next term we simply "add" -5 to the current one, and we know the first term is 12, ok, so,

$\bf n^{th}\textit{ term of an arithmetic sequence} \\\\ a_n=a_1+(n-1)d\qquad \begin{cases} n=n^{th}\ term\\ a_1=\textit{first term's value}\\ d=\textit{common difference}\\ ----------\\ d=-5\\ a_1=12\\ n=33 \end{cases} \\\\\\ a_{33}=12+(33-1)(-5)\implies a_{33}=12+(32)(-5) \\\\\\ a_{33}=12-160\implies a_{33}=-148$