Question

In the following exercise, determine if the sequence is geometric, and if so, indicate the common ratio.

$3,12,48,192,768,3072,...$

Short answer

Yes, the sequence is geometric with common ratio equal to$4$.

Step-by-step solution

Step 1. Determine if the sequence is geometric.

Consider the sequence $3,12,48,192,768,3072,...$

The sequence is said to be geometric sequence if the ratio between the consecutive terms is equal.

role="math" localid="1645298916718" $\begin{array}{rcl}\frac{12}{3}& =& 4\\ \frac{48}{12}& =& 4\\ \frac{192}{48}& =& 4\\ \frac{768}{192}& =& 4\\ \frac{3072}{768}& =& 4\end{array}$

As the ratio is constant between the consecutive terms, the sequence is a geometric sequence.

Find the common ratio.

As $\begin{array}{rcl}& & \frac{12}{3}\end{array}=\begin{array}{rcl}& & \frac{48}{12}\end{array}=\begin{array}{rcl}& & \frac{192}{48}\end{array}\begin{array}{rcl}& =& \end{array}\begin{array}{rcl}& & \frac{768}{192}\end{array}\begin{array}{rcl}& =& \end{array}\begin{array}{rcl}& & \frac{3072}{768}\end{array}\begin{array}{rcl}& =& \end{array}\begin{array}{rcl}& & 4\end{array}$, common ratio is$4$.