If a variable takes values \[a,ar,a{{r}^{2}},......,a{{r}^{n-1}}\], which of the relations between means hold?
(a)\[AH={{G}^{2}}\]
(b)\[\dfrac{A+H}{2}=G\]
(c)\[A>G>H\]
(d)\[A=G=H\]

Hint: Find the relationships connecting arithmetic, Geometric and harmonic mean with respect to their formulas. Find it using the basic concepts of AP, GP and HP.

Complete step-by-step answer:

To find this we need to know about the relationship between Arithmetic progression, Geometric progression and Harmonic progression.

Arithmetic terms are the sequence of numbers in which the difference between any 2 adjacent terms are constant and they are denoted as d i.e., common difference.

\[d={{a}_{2}}-{{a}_{1}}={{a}_{3}}-{{a}_{1}}...\]

Geometric progression are the terms in which if the first term is known and the common ratio is known then the next term in series can be calculated by multiplying the first term to common ratio (r).

\[r=\dfrac{{{a}_{2}}}{{{a}_{1}}}=\dfrac{{{a}_{3}}}{{{a}_{2}}}.....\]

Harmonic progression is the reciprocal of the values of terms in Arithmetic progression. If an AP is a, b, c, d, e then HP is its reciprocal.

i.e. \[\dfrac{1}{a},\dfrac{1}{b},\dfrac{1}{c},\dfrac{1}{d},\dfrac{1}{e}\]

We are given the series \[a,ar,a{{r}^{2}},......,a{{r}^{n-1}}\].

If a, b are in HP, then the harmonic mean is represented as,

\[H=\dfrac{2ab}{a+b}\]

If AM, GM and HM are the arithmetic, geometric and harmonic means between a and b, then we get the results,

\[AM=\dfrac{a+b}{2}\], arithmetic mean

\[GM=\sqrt{ab}\], geometric mean

\[HM=\dfrac{2ab}{a+b}\], harmonic mean

If we find, \[AM\times HM=\left( \dfrac{a+b}{2} \right)\left( \dfrac{2ab}{a+b} \right)=ab\].

\[\begin{align}

  & AM\times HM=ab \\

 & GM=\sqrt{ab} \\

\end{align}\]

Taking the square root of \[AM\times HM\] we get, \[\sqrt{ab}\].

\[\therefore GM=\sqrt{AM\times HM}\]

Squaring on both sides, we get

\[G{{M}^{2}}=AM\times HM\]

From this it's clear that,

\[{{G}^{2}}=AH\]


\[\therefore \] Option (a) is the correct answer.


Note: We can say that the arithmetic, geometric and harmonic means between any 2 positive quantities i.e. like a and b are in descending order of magnitude.

i.e. \[AM-GM=\dfrac{a+b}{2}-\sqrt{ab}=\dfrac{a+b-2\sqrt{ab}}{2}={{\left( \dfrac{\sqrt{a}-\sqrt{b}}{\sqrt{2}} \right)}^{2}}\]

\[\therefore AM>GM>HM\]