Chapter 8.4, Problem 36E

To determine

Check whether the given series is arithmetic, geometric or neither.

Answer to Problem 36E

Thegiven series is a geometric series.

Explanation of Solution

Given:

The given series is $13.5,40.5,121.5,364.5,\mathrm{...}$

Calculation:

The general rule for the $n\text{th}$ term is $a_n=a_1+\left(n-1\right)d$ .

Where $a_1\to$ first term, $d\to$ common difference, $n\to$ number of terms.

In the given sequence $13.5,40.5,121.5,364.5,\mathrm{...}$

  $\begin{array}{l}{a}_2-a_1=40.5-13.5=27\\ a_3-a_2=121.5-40.5=81\end{array}$

The common difference is not constant.

In a geometric sequence, the ratio of any term to the previous term, called the common ratio, is constant.

Find the common ratio of any term to the previous term of the sequence $13.5,40.5,121.5,364.5,\mathrm{...}$

  $\begin{array}{l}\frac{a_2}{a_1}=\frac{40.5}{13.5}=3\\ \frac{a_3}{a_2}=\frac{121.5}{40.5}=3\\ \frac{a_4}{a_3}=\frac{364.5}{121.5}=3\end{array}$

The common ratio $\left(3\right)$ is constant.

Hence the given series is a geometric series.