Chapter 12.3, Problem 18E

To determine

Find whether the sequence is geometric .If it is then, find the common ratio.

Answer to Problem 18E

Yes, the terms are in geometric sequence and the common ratio is

  $e^2$ .

Explanation of Solution

Given:

It is given in the question that the term are $e^2,e^4,e^6,e^8,\mathrm{............}$ .

Concept Used:

In this, use the concept that divide every two consecutive terms, if the constant term is arriving in all then the term are in geometric and the constant term is the common difference.

Calculation:

Here, the term are $e^2,e^4,e^6,e^8,\mathrm{............}$ .

Divide every two consecutive terms,

  $\begin{array}{l}{e}^4÷e^2=e^2\\ e^6÷e^4=e^2\\ e^8÷e^6=e^2\end{array}$

After dividing it is seen that the same constant term occurred in each , so these can be the terms of an geometric sequence.

Now, to find the common ratio r , we divide two consecutive terms

  $e^4÷e^2=e^2$

Conclusion:

Yes, the terms are in geometric sequence and the common ratio is $e^2$ .