Question

for each sequence, determine whether it appears to be geometric. if it does, find the common ratio. (a) 8, 6, 4, 2, ... geometric common ratio: r = not geometric (b) -10, -40, -160, -640, ... geometric common ratio: r = not geometric (c) 80, 40, 20, 10, ... geometric common ratio: r = not geometric

Explanation:

Step1: Analyze sequence (a)

A geometric sequence has a common ratio \( r = \frac{a_{n + 1}}{a_n} \). For \( 8, 6, 4, 2, \dots \), check ratios: \( \frac{6}{8}=\frac{3}{4} \), \( \frac{4}{6}=\frac{2}{3} \). Ratios not equal, so not geometric.

Step2: Analyze sequence (b)

For \( -10, -40, -160, -640, \dots \), calculate \( r \): \( \frac{-40}{-10}=4 \), \( \frac{-160}{-40}=4 \), \( \frac{-640}{-160}=4 \). Common ratio \( r = 4 \), so geometric.

Step3: Analyze sequence (c)

For \( 80, 40, 20, 10, \dots \), calculate \( r \): \( \frac{40}{80}=\frac{1}{2} \), \( \frac{20}{40}=\frac{1}{2} \), \( \frac{10}{20}=\frac{1}{2} \). Common ratio \( r=\frac{1}{2} \), so geometric.

Answer:

(a) Not geometric
(b) Geometric, Common ratio: \( r = 4 \)
(c) Geometric, Common ratio: \( r=\frac{1}{2} \)