Question

for each sequence, determine whether it appears to be geometric.
if it does, find the common ratio.
(a) 256, -64, 16, -4, ...
geometric
common ratio: r =
not geometric
(b) 2, 10, 50, 250, ...
geometric
common ratio: r =
not geometric
(c) 6, 9, 12, 15, ...
geometric
common ratio: r =
not geometric

Explanation:

Step1: Analyze sequence (a)

A geometric sequence has a common ratio \( r \) where \( r=\frac{a_{n + 1}}{a_n} \) for all \( n \). For sequence \( 256,-64,16,-4,\dots \):
\( r_1=\frac{-64}{256}=-\frac{1}{4} \)
\( r_2=\frac{16}{-64}=-\frac{1}{4} \)
\( r_3=\frac{-4}{16}=-\frac{1}{4} \)
Since the ratio is constant, it is geometric with \( r = -\frac{1}{4} \).

Step2: Analyze sequence (b)

For sequence \( 2,10,50,250,\dots \):
\( r_1=\frac{10}{2}=5 \)
\( r_2=\frac{50}{10}=5 \)
\( r_3=\frac{250}{50}=5 \)
Since the ratio is constant, it is geometric with \( r = 5 \).

Step3: Analyze sequence (c)

For sequence \( 6,9,12,15,\dots \):
\( r_1=\frac{9}{6}=\frac{3}{2} \)
\( r_2=\frac{12}{9}=\frac{4}{3} \)
Since \( \frac{3}{2}
eq\frac{4}{3} \), the ratio is not constant, so it is not geometric.

Answer:

(a) Geometric, Common ratio: \( r = -\frac{1}{4} \)
(b) Geometric, Common ratio: \( r = 5 \)
(c) Not geometric