Question

Determine if each sequence is geometric. If so indicate the common ratio.

ⓐ.$7,21,63,189,567,1,701,\dots$

ⓑ.$64,16,4,1,14,116,\dots$

ⓒ.$2,4,12,48,240,1,440,\dots$

Short answer

ⓐ. Yes, the sequence is geometric with common ratio equal to $3$.

ⓑ. Yes, the sequence is geometric with common ratio equal to $\frac{1}{4}$

ⓒ. No, the sequence is not geometric.

Step-by-step solution

Step 1. Given information

Sequences are as follows:

ⓐ$7,21,63,189,567,1,701,\dots$

ⓑ$64,16,4,1,14,116,\dots$

ⓒ$2,4,12,48,240,1,440,\dots$

Step 2. Determine if each sequence is geometric. If so indicate the common ratio. 

A sequence is said to be geometric if ratio of the consecutive terms is same.

ⓐ $7,21,63,189,567,1,701,...$

role="math" localid="1645264969253" $\begin{array}{rcl}\frac{21}{7}& =& 3\\ \frac{63}{21}& =& 3\\ \frac{189}{63}& =& 3\\ \frac{567}{189}& =& 3\\ \frac{1701}{567}& =& 3\end{array}$

So, the sequence is geometric and common ratio is equal to $3$

Step 3. Determine if each sequence is geometric. If so indicate the common ratio. 

ⓑ $64,16,4,1,\frac{1}{4},\frac{1}{16},...$

$$\begin{gathered} \frac{16}{64}=\frac{1}{4} \\ \frac{4}{16}=\frac{1}{4} \\ \frac{1}{4}=\frac{1}{4} \\ \frac{{\displaystyle \frac{1}{4}}}{1}=\frac{1}{4} \\ \frac{{\displaystyle \frac{1}{16}}}{{\displaystyle \frac{1}{4}}}=\frac{1}{4} \end{gathered}$$

So, the ratio of consecutive terms is same.

Yes, the sequence is geometric with common ratio equal to$\frac{1}{4}$

Step 4. Determine if each sequence is geometric. If so indicate the common ratio. 

ⓒ $2,4,12,48,240,1,440,...$

role="math" $\begin{array}{rcl}\frac{4}{2}& =& 2\\ \frac{12}{4}& =& 3\end{array}$

As $\frac{4}{2}\ne \frac{12}{4}$, the sequence is not geometric.