Question

Find the thirtieth term of the following sequence. -6, -4, -2, 0, ...

Answer 1

So hmm -6, -4, -2, 0?  what the heck is going on?

well, from -6 to -4, is really a +2 "difference", and from -4 to -2 is the same amount.  So to get the next term's value, you simply "add 2", therefore, 2 is the "common difference" in this arithmetic sequence.

let's also notice that -6 is our first fellow.

$\bf n^{th}\textit{ term of an arithmetic sequence}\\\\ a_n=a_1+(n-1)d\qquad \begin{cases} n=n^{th}\ term\\ a_1=\textit{first term's value}\\ d=\textit{common difference}\\ ----------\\ d=2\\ a_1=-6\\ n=13 \end{cases} \\\\\\ a_n=-6+(13-1)2\implies a_{13}=-6+(13-1)2 \\\\\\ a_{13}=-6+(12)2\implies a_{13}=-6+24\implies a_{13}=18$