Chapter 8.3, Problem 14E

To determine

To find whether the sequence $5,1,0.2,0.04,...$ is geometric and if so, then find its common ratio.

Answer to Problem 14E

The sequence $5,1,0.2,0.04,...$ is geometric and the common ratio $r=\frac{1}{5}$ .

Explanation of Solution

Given information :

Let the sequence be $5,1,0.2,0.04,...$ .

Concept used:

In a sequence $a_1,a_2,a_3,a_4,\dots \dots ,a_n,\dots$ if ratio of the consecutive terms is the same then it is defined as Geometric Sequence.

So, a number r is defined as the common ratio of the geometric sequence if $\frac{a_2}{a_1}=\frac{a_3}{a_2}=\frac{a_4}{a_3}=\dots \dots =r.$

Calculation:

Here, $a_1=5,a_2=1,a_3=0.2,a_4=0.04,\dots$

Calculate the ratio of the consecutive terms:

  $\begin{array}{l}\frac{a_2}{a_1}=\frac{1}{5}=\frac{1}{5}\\ \frac{a_3}{a_2}=\frac{0.2}{1}=\frac{2}{10}=\frac{1}{5}\\ \frac{a_4}{a_3}=\frac{0.04}{0.2}=\frac{4\times 10}{2\times 100}=\frac{1}{5}\\ ⋮\end{array}$

And so on.

Since, $\frac{a_2}{a_1}=\frac{a_3}{a_2}=\frac{a_4}{a_3}=\dots \dots =r=\frac{1}{5}$ , therefore the sequence is geometric.