Chapter SH, Problem 7.31EP

To determine

To explain whether sequence is arithmetic geometric or neither.

Answer to Problem 7.31EP

Sequence is not arithmetic and not geometric.

Explanation of Solution

Given information:

The sequence

  $100,90,85,75,\mathrm{60........}$

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

  $100,90,85,75,\mathrm{60........}$

  $a_1=100$

  $a_2=90$

  $a_3=85$.

  .

  .

And so on.

Check for arithmetic sequence

Using common difference formula

Common difference is for first two terms

  $d=90-100$

On solving,

  $d=-10$

Common difference for second and third term

  $d=85-90$

On solving,

  $d=-5$

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

Sequence is not arithmetic

Check for geometric

Using common ratio formula

Common ratio for first two terms

  $r=\frac{90}{100}$

On solving,

  $r=0.9$

Common ratio for second and third terms

  $r=\frac{85}{90}$

On solving,

  $r=0.94$

As shown above common ratio of the terms is not same

Then, sequence is not geometric

Hence, sequence is not arithmetic and not geometric.