Chapter 11, Problem 1CT

To determine

To identify:

The sequence is arithmetic or geometric or neither $\frac{2}{3},\frac{3}{5},\frac{4}{7},\frac{5}{9},\mathrm{...}$

Answer to Problem 1CT

The sequence is neither geometric nor arithmetic.

Explanation of Solution

Given information:

  $\begin{array}{l}{a}_1=\frac{2}{3}\\ a_2=\frac{3}{5}\\ a_3=\frac{4}{7}\\ a_4=\frac{5}{9}\end{array}$

Calculation:

Consider the sequence:

Check using arithmetic sequence:

  $\begin{array}{l}d=a_2-a_1=\frac{3}{5}-\frac{2}{3}=-\frac{1}{15}\\ d=a_3-a_2=\frac{4}{7}-\frac{3}{5}=-\frac{1}{35}\\ d=a_4-a_3=\frac{5}{9}-\frac{4}{7}=-\frac{1}{63}\end{array}$

This sequence is not arithmetic sequence.

Check using geometric sequence:

  $\begin{array}{l}r=\frac{a_2}{a_1}=\frac{3/5}{2/3}=0.9\\ r=\frac{a_3}{a_2}=\frac{4/7}{3/5}=0.9523\\ r=\frac{a_4}{a_3}=\frac{5/9}{4/7}=0.9722\end{array}$

This sequence is not geometric sequence.

The sequence is neither geometric nor arithmetic.