Question

which answers are geometric sequences? there is more than one correct answer. select all that apply. {2.5,4,5.5,7,...} {-13,-6,1,8,...} {1,-1,1,-1,1,...} {0,10,20,30,...} {2,-10,50,-250,...} {108,36,12,4,...}

Explanation:

Step1: Recall geometric - sequence definition

A geometric sequence has a common ratio $r=\frac{a_{n + 1}}{a_n}$ for $n\geq1$.

Step2: Check sequence $\{2.5,4,5.5,7,\cdots\}$

$\frac{4}{2.5}=1.6$, $\frac{5.5}{4}=1.375$. No common ratio.

Step3: Check sequence $\{-13,-6,1,8,\cdots\}$

$\frac{-6}{-13}=\frac{6}{13}$, $\frac{1}{-6}=-\frac{1}{6}$. No common ratio.

Step4: Check sequence $\{1, - 1,1,-1,1,\cdots\}$

$\frac{-1}{1}=-1$, $\frac{1}{-1}=-1$, $\frac{-1}{1}=-1$. Common ratio $r = - 1$.

Step5: Check sequence $\{0,10,20,30,\cdots\}$

$\frac{10}{0}$ is undefined. Not a geometric sequence.

Step6: Check sequence $\{2,-10,50,-250,\cdots\}$

$\frac{-10}{2}=-5$, $\frac{50}{-10}=-5$, $\frac{-250}{50}=-5$. Common ratio $r=-5$.

Step7: Check sequence $\{108,36,12,4,\cdots\}$

$\frac{36}{108}=\frac{1}{3}$, $\frac{12}{36}=\frac{1}{3}$, $\frac{4}{12}=\frac{1}{3}$. Common ratio $r=\frac{1}{3}$.

Answer:

$\{1,-1,1,-1,1,\cdots\}$, $\{2,-10,50,-250,\cdots\}$, $\{108,36,12,4,\cdots\}$