Unit-I: Sets and Functions
Unit-II: Algebra
Unit-III: Coordinate Geometry
Unit-IV: Calculus
Unit-V: Mathematical Reasoning
Unit-VI: Statistics and Probability

Relation between A.M, G.M and H.M

If  a and  b be two positive numbers then their Arithmetic mean ‘A’, Geometric mean ‘G’, and Harmonic mean ‘H’ are given by

 A=\dfrac{a+b}{2},\ \text{G}=\sqrt{ab} and  \text{H}=\dfrac{2ab}{a+b}

Then  \text{A-G}=\dfrac{a+b}{2}-\sqrt{ab}=\dfrac{1}{2}(a+b-2\sqrt{ab})

 =\dfrac{1}{2}\left(\sqrt{a}-(\sqrt{b})\ ^{2} \right)\geq 0

 \therefore A\geq G

\begin{aligned}\text{ Also } \text{G-H}&=\sqrt{ab}-\dfrac{2ab}{a+b} \\&= \sqrt{\dfrac{ab}{a+b}} \left( a+b-2\sqrt{ab} \right)\\&=\sqrt{\dfrac{ab}{a+b}} \left( \sqrt{a} -\sqrt{b}\right)^{2}\geq 0 \end{aligned}

 \therefore \text{G}\geq \text{H}

Hence, we get  \text{A}\geq \text{G}\geq \text{H}

Note -:

If  \text{A, G, H} are respectively Arithmetic, Geometric and Harmonic mean of  a_1,a_2...,a_n then  \text{A}\geq \text{G}\geq \text{H}

Scroll to Top