Chapter 7.8, Problem 6P

Determine whether the sequence is a geometric sequence. Explain.

$25,35,45,55,\dots$

To determine

whether the sequence $25,35,45,55,...$ is geometric.

Answer to Problem 6P

The sequence $25,35,45,55,...$ is not geometric.

Explanation of Solution

Given information :

The sequence is $25,35,45,55,...$ .

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, $a_1=25,a_2=35,a_3=45,a_4=55,\dots$

Calculate the ratio of the consecutive terms:

  $\begin{array}{l}\frac{a_2}{a_1}=\frac{35}{25}=\frac{7}{5}\\ \frac{a_3}{a_2}=\frac{45}{35}=\frac{9}{7}\\ \frac{a_4}{a_3}=\frac{55}{45}=\frac{11}{9}\\ ⋮\end{array}$

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.