Chapter 11.1, Problem 7WE

To determine

To calculate:

The following sequence is arithmetic, geometric or neither and the missing terms:

  $1,\frac{1}{4},\frac{1}{9},\frac{1}{16},?,?$

Answer to Problem 7WE

The sequence $\left(1,\frac{1}{4},\frac{1}{9},\frac{1}{16},?,?\right)$ is neither a geometric sequence nor arithmetic sequence.

Explanation of Solution

Given information:

  $1,\frac{1}{4},\frac{1}{9},\frac{1}{16},?,?$

Calculation:

Check the common difference first to categorize the given sequence into arithmetic, geometric or neither:

  $\begin{array}{l}{a}_1=1\\ a_2=\frac{1}{4}\\ a_3=\frac{1}{9}\\ a_4=\frac{1}{16}\end{array}$

Now, the common difference is calculated as:

  $\begin{array}{l}r=a_2-a_1=\frac{1}{4}-1=\frac{-3}{4}\\ r=a_3-a_2=\frac{1}{9}-\frac{1}{4}=\frac{-5}{36}\\ r=a_4-a_3=\frac{1}{16}-\frac{1}{9}=\frac{-7}{144}\end{array}$

Therefore, the given sequence is not arithmetic sequence as the common ratio is not same.

Now, check for geometric sequence:

  $\begin{array}{l}r=\frac{a_2}{a_1}=\frac{\left(\frac{1}{4}\right)}{1}=\frac{1}{4}\\ r=\frac{a_3}{a_2}=\frac{\left(\frac{1}{9}\right)}{\left(\frac{1}{4}\right)}=\frac{4}{9}\\ r=\frac{a_4}{a_3}=\frac{\left(\frac{1}{16}\right)}{\left(\frac{1}{9}\right)}=\frac{9}{16}\end{array}$

Here, the given sequence is not a geometric sequence as the common ratio is not same.

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