JEE Main & Advanced Mathematics Sequence & Series Properties of Arithmetic, Geometric, Harmonic Means Between Two Given Numbers

Let \[A,\,\,G\] and \[H\] be arithmetic, geometric and harmonic means of two numbers \[a\] and \[b\].

Then, \[A=\frac{a+b}{2},\,G=\sqrt{ab}\] and \[H=\frac{2ab}{a+b}\].

These three means possess the following properties :

(1) \[A\ge G\ge H\]

\[A=\frac{a+b}{2},\,G=\sqrt{ab}\] and \[H=\frac{2ab}{a+b}\]

\[\therefore \] \[A-G=\frac{a+b}{2}-\sqrt{ab}=\frac{{{(\sqrt{a}-\sqrt{b})}^{2}}}{2}\ge 0\]\[\Rightarrow \] \[A\ge G\]   …..(i)

\[G-H=\sqrt{ab}-\frac{2ab}{a+b}=\sqrt{ab}\left( \frac{a+b-2\sqrt{ab}}{a+b} \right)=\frac{\sqrt{ab}}{a+b}{{(\sqrt{a}-\sqrt{b})}^{2}}\ge 0\]

\[\Rightarrow \] \[G\ge H\]                                                                     …..(ii)

From (i) and (ii), we get \[A\ge G\ge H\].

Note that the equality holds only when \[a=b\].

(2) \[A,\,\,G,\,\,H\] from a G.P., i.e., \[{{G}^{2}}=AH\]

\[AH=\frac{a+b}{2}\times \frac{2ab}{a+b}=ab={{(\sqrt{ab})}^{2}}={{G}^{2}}\]. Hence, \[{{G}^{2}}=AH\]

(3) The equation having \[a\] and \[b\] as its roots is

\[{{x}^{2}}-2Ax+{{G}^{2}}=0\]

The equation having \[a\] and \[b\] its roots is

\[{{x}^{2}}-(a+b)x+ab=0\]

\[\Rightarrow \] \[{{x}^{2}}-2Ax+{{G}^{2}}=0\],     \[\left[ \because A=\frac{a+b}{2}\text{ and }G=\sqrt{ab} \right]\].

The roots \[a,\,\,\,b\] are given by \[A\pm \sqrt{{{A}^{2}}-{{G}^{2}}}\].

(4) If \[A,\,\,G,\,\,H\] re arithmetic, geometric and harmonic means between three given numbers \[a,\,\,b\] and \[c,\] then the equation having \[a,\,\,b,\,\,c\] as its roots is \[{{x}^{3}}-3A{{x}^{2}}+\frac{3{{G}^{3}}}{H}x-{{G}^{3}}=0\]

where \[A=\frac{a+b+c}{3},\,G={{(abc)}^{1/3}}\] and \[\frac{1}{H}=\frac{\frac{1}{a}+\frac{1}{b}+\frac{1}{c}}{3}\]

\[\Rightarrow \] \[a+b+c=3A,\,abc={{G}^{3}}\] and \[\frac{3{{G}^{3}}}{H}=ab+bc+ca\]

The equation having \[a,\,\,b,\,\,c\] as its roots is

\[{{x}^{3}}-(a+b+c){{x}^{2}}+(ab+bc+ca)x-abc=0\]

\[\Rightarrow \] \[{{x}^{3}}-3A{{x}^{2}}+\frac{3{{G}^{3}}}{H}x-{{G}^{3}}=0\].