Chapter 8.3, Problem 16E

To determine

To find whether the sequence $\frac{1}{5},\frac{2}{7},\frac{3}{9},\frac{4}{11},...$ is geometric and if so, then find its common ratio.

Answer to Problem 16E

The sequence $\frac{1}{5},\frac{2}{7},\frac{3}{9},\frac{4}{11},...$ is not geometric.

Explanation of Solution

Given information :

Let the sequence be $\frac{1}{5},\frac{2}{7},\frac{3}{9},\frac{4}{11},...$ .

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, $a_1=\frac{1}{5},a_2=\frac{2}{7},a_3=\frac{3}{9},a_4=\frac{4}{11},\dots$

Calculate the ratio of the consecutive terms:

  $\begin{array}{l}\frac{a_2}{a_1}=\frac{\frac{2}{7}}{\frac{1}{5}}=\frac{2}{7}\times 5=\frac{10}{7}\\ \frac{a_3}{a_2}=\frac{\frac{3}{9}}{\frac{2}{7}}=\frac{3}{9}\times \frac{7}{2}=\frac{7}{6}\\ \frac{a_4}{a_3}=\frac{\frac{4}{11}}{\frac{3}{9}}=\frac{4}{11}\times \frac{9}{3}=\frac{12}{11}\\ ⋮\end{array}$

And so on.

Since, $\frac{a_2}{a_1}\ne \frac{a_3}{a_2}\ne \frac{a_4}{a_3}\ne \dots \dots$ , therefore the sequence is not geometric.