Answer:
a. We can tell if there will be more than one solution for the geometric means if the set of numbers is an even number.
b. This happens because, there are two possible values for the nth root of a set of even numbers.
Explanation:
a. How can you tell if there will be more than one solution for the geometric means?
We can tell if there will be more than one solution for the geometric means if the set of numbers is an even number.
Since geometric means of n numbers from x₁,x₂, x₃,...,xₙ is
GM = ⁿ√x₁,x₂, x₃,...,xₙ .
Now, if n is an even number, the geometric means would have both a positive and negative value.
For example, if n = 2,
GM = ±√x₁,x₂ which is
GM = +√x₁,x₂ or -√x₁,x₂.
b. Why does this happen with the geometric means?
This happens because, there are two possible values for the nth root of a set of even numbers. For example if n = 2,
GM = ±√x₁,x₂ which is
GM = +√x₁,x₂ or -√x₁,x₂.
So, we have two possible values for the geometric means.