Question

How do you determine whether the sequence is $ 80,40,20,10,5,... $ is arithmetic and if it is, what is the common difference?

Answer 1

Hint: A sequence is said to be arithmetic or geometric if the difference or ratio of two consecutive terms is the same for the entire sequence. In order to solve the given question, check whether the difference between all the consecutive terms is the same.

Complete step-by-step answer:
Given to us is a sequence $ 80,40,20,10,5,... $
In order to check whether this sequence is arithmetic, we have to first calculate the common difference. The common difference of a sequence is given as the difference between two consecutive terms.
Let us first find the difference between the first term and the second,
$ d = 40 - 80 = - 40 $
Now, the difference between the third and the second term,
 $ d = 20 - 40 = - 20 $
Now, the difference between the fourth and the third term,
 $ d = 10 - 20 = - 10 $
Similarly, the difference between the fifth and the fourth term,
 $ d = 5 - 10 = - 5 $
Clearly, the difference between the consecutive terms is not the same which means that the given sequence is not an arithmetic sequence or progression.
We can check whether the given sequence is a geometric sequence in a similar fashion. First let us find the ratio between the first and the second term
 $ r = \dfrac{{40}}{{80}} = \dfrac{1}{2} $
The ratio between the second and the third term is
 $ r = \dfrac{{20}}{{40}} = \dfrac{1}{2} $
The ratio between the third and the fourth term is
 $ r = \dfrac{{10}}{{20}} = \dfrac{1}{2} $
We can see that the ratios of all the consecutive terms are equal to $ \dfrac{1}{2} $ meaning that the given sequence has a common ratio and is a geometric progression.

Note: It is to be noted that the formula for the common difference of a sequence is $ {a_2} - {a_1} $ or $ {a_3} - {a_2} $ or $ {a_4} - {a_3} $ and so on. Here the second term comes before the first or the third term comes before the second and so on and never the other way around.