Question

How many terms are in the arithmetic sequence 7, 0, −7, . . . , −175?

Answer 1

So hmmm 7, 0, -7.... what the dickens is going on?   hmmm is really dropping each time by 7, so, 7-7, 0, and 0-7, -7 and so on.

so, the "common difference" is then -7, and our first term is 7, now, who's -175?  let's check.

$\bf a_n=a_1+(n-1)d\qquad \begin{cases} n=n^{th}\ term\\ a_1=\textit{first term's value}\\ d=\textit{common difference}\\ ----------\\ d=-7\\ a_1=7\\ a_n=-175 \end{cases} \\\\\\ -175=7+(n-1)(-7)\implies -175=7+7-7n \\\\\\ 7n=14+175\implies 7n=189\implies n=\cfrac{189}{7}\implies n=\stackrel{terms}{27}$