Chapter 8.5, Problem 108E

To determine

To check: if the given sequence is arithmetic, geometric or neither and explain

Answer to Problem 108E

Neither arithmetic nor geometric sequence.

Explanation of Solution

Given information: We have given a sequence $4,5,9,14,23,\mathrm{...}$

First of all check the differences of next terms to previous terms of the given sequence.

  $\begin{array}{l}{d}_1=a_2-a_1=5-4=1\\ d_2=a_3-a_2=9-5=4\\ d_3=a_4-a_3=14-9=5\\ d_4=a_5-a_4=23-14=9\end{array}$

We could see that differences of next terms to previous terms are not same. Therefore, it’s not an Arithmetic sequence.

Now, we need to check for the ratios of next terms to previous terms to determine if it is a geometric sequence.

  $\begin{array}{l}{r}_1=\frac{a_2}{a_1}=\frac{5}{4}\\ r_2=\frac{a_3}{a_2}=\frac{9}{5}\\ r_3=\frac{a_4}{a_3}=\frac{14}{9}\\ r_4=\frac{a_5}{a_4}=\frac{23}{14}\end{array}$

We could see that ratios of next terms to previous terms are not same. So, it’s not a Geometric sequence.

Therefore, the given sequence is neither arithmetic nor geometric sequence.