Geometric Sequences II

Problem 224: Consider the sequence $b = \{6,3,3/2,3/4,3/8,\dots \}$

a. What is the common ration?

b. What are the next five terms in the sequence?

Solution: A geometric sequence is different from an arithmetic sequence. In an arithmetic sequence, we look for a common difference between the terms in the sequence whereas in the geometric sequence we look for a common ratio between the terms of the sequence.

a. Note that

(1) $\begin{align*}\frac{3}{6} & = \frac{1}{2} \\ \frac{\frac{3}{2}}{3}& =\frac{1}{2} \\ \frac{\frac{3}{4}}{\frac{3}{2}} & = \frac{1}{2} \\ \frac{\frac{3}{8}}{\frac{3}{4}} & = \frac{1}{2}. \end{align*}$

This means that the common ratio between the terms is $\frac{1}{2}$ .

b. Using the common ratios we can get the next five terms.

(2) $\begin{align*} \frac{3}{8} \cdot \frac{1}{2} & = \frac{3}{16} \\ \frac{3}{16} \cdot \frac{1}{2} & = \frac{3}{32} \\ \frac{3}{32} \cdot \frac{1}{2} & = \frac{3}{64} \\ \frac{3}{64} \cdot \frac{1}{2} & = \frac{3}{128} \\ \frac{3}{128} \cdot \frac{1}{2} & = \frac{3}{256}\end{align*}$

Leave a Reply Cancel reply