Question

How do you determine whether the sequence $ 1, - \dfrac{1}{2},\dfrac{1}{4}, - \dfrac{1}{8},.... $ is a geometric and is it is, what is the common ratio?

Answer 1

Hint: To solve this type of problem we should know about basic term like;
Geometric series: it is a sequence in mathematics in which the previous term is multiplied by a constant term which is called a common ratio. Common ratio is a non-zero number.
Its general form is, $ a,an,a{n^2},a{n^3},........... $ where $ a $ is first term $ n $ is common ratio.
I.e. $ 1,2,4,8,16,....... $ here common ratio is $ 2 $ and previous term is multiplied by $ 2 $ to get next term.

Complete step by step solution:
As given series is,
  $ 1, - \dfrac{1}{2},\dfrac{1}{4}, - \dfrac{1}{8},.... $
To check is it a geometric progression or not we will first check it.
By following the general format of geometric progression.
  $ a,an,a{n^2},a{n^3},........... $
So, we can calculate the common factor by dividing the preceding term by the previous one and if we get the same number then it will be in geometric progression.
Dividing second number by first number,
  $ \dfrac{{ - \dfrac{1}{2}}}{1} = - \dfrac{1}{2} $
Dividing third number by second number,
  $ \dfrac{{\dfrac{1}{4}}}{{ - \dfrac{1}{2}}} = - \dfrac{1}{2} $
Dividing fourth number by third number,
  $ \dfrac{{ - \dfrac{1}{8}}}{{\dfrac{1}{4}}} = - \dfrac{1}{2} $
As we get from the above calculation there is a common result for all above calculations.
So, it follows the format of geometric progression.
Hence, the first term is $ a = 1 $ and the common ratio is $ n = - \dfrac{1}{2} $ .

Note: Geometric progression is used in our daily life in various forms. It is used in mathematics, physics, finance, engineering, let for calculating decay of atoms or calculating radioactivity. Most influence use of it in calculating population growth as well as it is used in calculation compound interest like complex problems in the finance sector.