Question

which of the following are geometric sequences? check all that apply. a. -9, -3, -1, -1/3, -1/9, -1/27 b. 10, 5, 2.5, 1.25, 0.625, 0.3125 c. 5, 10, 15, 20, 25 d. 1, 1, 2, 3, 5, 8, 13

Explanation:

Step1: Recall geometric sequence definition

A geometric sequence has a common ratio \( r \), where \( r=\frac{a_{n + 1}}{a_n} \) is constant for all \( n \).

Step2: Analyze Option A

For sequence \(-9, -3, -1, -\frac{1}{3}, -\frac{1}{9}, -\frac{1}{27}\):
\( r=\frac{-3}{-9}=\frac{1}{3} \), \( \frac{-1}{-3}=\frac{1}{3} \), \( \frac{-\frac{1}{3}}{-1}=\frac{1}{3} \), etc. Ratio is constant (\( \frac{1}{3} \)), so it's geometric.

Step3: Analyze Option B

For sequence \( 10, 5, 2.5, 1.25, 0.625, 0.3125 \):
\( r=\frac{5}{10}=\frac{1}{2} \), \( \frac{2.5}{5}=\frac{1}{2} \), \( \frac{1.25}{2.5}=\frac{1}{2} \), etc. Ratio is constant (\( \frac{1}{2} \)), so it's geometric.

Step4: Analyze Option C

For sequence \( 5, 10, 15, 20, 25 \):
\( r=\frac{10}{5}=2 \), \( \frac{15}{10}=1.5 \). Ratios are not equal, so it's arithmetic (common difference 5), not geometric.

Step5: Analyze Option D

For sequence \( 1, 1, 2, 3, 5, 8, 13 \):
\( r=\frac{1}{1}=1 \), \( \frac{2}{1}=2 \). Ratios are not equal, it's a Fibonacci sequence (sum of previous two terms), not geometric.

Answer:

A. \(-9, -3, -1, -1/3, -1/9, -1/27\), B. \(10, 5, 2.5, 1.25, 0.625, 0.3125\)