Chapter 3.6, Problem 9PPS

To determine

To find:Whether the sequence $\frac{1}{2},\frac{3}{4},\frac{5}{8},\frac{7}{16},...$ is an arithmetic sequence.

Answer to Problem 9PPS

No.The sequence $\frac{1}{2},\frac{3}{4},\frac{5}{8},\frac{7}{16},...$ is not arithmetic.

Explanation of Solution

Given information:

The terms of the sequence aredefined as $a_1=\frac{1}{2},a_2=\frac{3}{4},a_3=\frac{5}{8},a_4=\frac{7}{16},\dots ,$

Concept used:

In a sequence $a_1,a_2,a_3,a_4,\dots \dots ,a_n,\dots$ if the difference between the consecutive terms are the same then it is defined as Arithmetic Sequence.

Here, $a_2-a_1=a_3-a_2=a_4-a_3=\dots \dots =d.$ The number $d$ is the common difference of the arithmetic sequence.

Calculations:

Calculate the difference of the consecutive terms:

  $\begin{array}{l}{a}_2-a_1=\frac{3}{4}-\frac{1}{2}=\frac{3-2}{4}=\frac{1}{4}\\ a_3-a_2=\frac{5}{8}-\frac{3}{4}=\frac{5-6}{8}=\frac{-1}{4}\\ a_4-a_3=\frac{7}{16}-\frac{5}{8}=\frac{7-10}{16}=\frac{-3}{16}\\ ⋮\end{array}$

And so on.

Since, the difference between the consecutive terms is not same, that is $a_2-a_1\ne a_3-a_2\ne a_4-a_3\dots$ , therefore the sequence is not arithmetic.