Chapter SH, Problem 7.30EP

To determine

To explain whether sequence is arithmetic geometric or neither.

Answer to Problem 7.30EP

Sequence is arithmetic but not geometric.

Explanation of Solution

Given information:

The sequence

  $-\frac{1}{2},-\frac{1}{4},0,\frac{1}{4},\frac{1}{2}\mathrm{.......}$

Formula used:

  $d=a_{next}-a_{present}$

d is common difference of the arithmetic sequence

  $r=\frac{a_{next}}{a_{present}}$

r is common ratio of geometric sequence

  $a_{next}$ is next term

  $a_{present}$ is present term

Calculation:

The sequence

  $-\frac{1}{2},-\frac{1}{4},0,\frac{1}{4},\frac{1}{2}\mathrm{.......}$

  $a_1=-\frac{1}{2}$

  $a_2=-\frac{1}{4}$

  $a_3=0$.

And so on.

Check for arithmetic sequence

Using common difference formula

Common difference is for first two terms

  $d=-\frac{1}{4}-\left(-\frac{1}{2}\right)$

On solving,

  $d=-\frac{2}{8}+\frac{4}{8}$

Then,

  

Common difference for second and third term

  $d=0-(-\frac{1}{4})$

On solving,

  $d=\frac{1}{4}$

As shown above common difference of all the term is same

Sequence is arithmetic

Check for geometric

Using common ratio formula

Common ratio for first two terms

  $r=\frac{-\frac{1}{4}}{-\frac{1}{2}}$

On solving,

  $r=\frac{1}{2}$

Common ratio for second and third terms

  $r=\frac{0}{-\frac{1}{4}}$

On solving,

  $r=0$

As shown above common ratio of all the term is not same

Then, sequence is not geometric

Hence, sequence is arithmetic but not geometric.