Question

prove that for any two positive numbers, the three means AM,GM and HM are in geometric progression

Answer 1

Step-by-step explanation:

In Maths, when we learn about sequences, we also come across the relation between AM, GM and HM. These three are average or mean of the respective series. AM stands for Arithmetic Mean, GM stands for Geometric Mean, and HM stands for Harmonic Mean. AM, GM and HM are the mean of Arithmetic Progression (AP), Geometric Progression (GP) and Harmonic Progression (HP) respectively. Before learning about the relationship between them, one should know about these three means along with their formulas

Answer 2

Answer:

Let a and b be any 2 positive real numbers

AM = $\frac{a+b}{2}$

GM = $\sqrt{ab}$

HM = $\frac{2ab}{a+b}$

Now, AM × HM = $\frac{a+b}{2}$ × $\frac{2ab}{a+b}$

                         = ab

                         = $(\sqrt{ab})^{2}$

                          = GM²

Thus, AM × HM = GM²

And, AM, GM and HM are in GP