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**Hint**: From the question student should understand that this sum is an application of formulae related to Arithmetic Mean , Geometric Mean , Harmonic Mean. First step towards solving this sum is noting down the formulae for sum upto $ n $ terms . Bring it in the simplest possible form in the next step. After this the student should remove the common terms and bring the relation between these means.

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__Complete step-by-step answer__In order to solve the numerical first step is to list down the formulae for Arithmetic Mean , Geometric Mean & Harmonic Mean.

$ {G_k} = {({a_1} \times {a_2} \times {a_3}.........{a_k})^{1/k}}..............(1) $

Where $ k $ is the last term of the expression.

We can simplify equation $ 1 $ as below

$ {G_k} = {({a_1}r)^{\dfrac{{k - 1}}{2}}}..............(2) $

Following is the formula for Arithmetic progression upto $ k $ terms.

$ {A_k} = \dfrac{{{a_1} + {a_2} + ......{a_k}}}{k}..........(3) $

$ {A_k} = \dfrac{{{a_1}(1 + r + .......{r^{k - 1}})}}{k}..........(4) $

$ {A_k} = \dfrac{{{a_1}({r^k} - 1)}}{{(r - 1)k}}..........(5) $

Noting down the formula for Harmonic Progression upto $ k $ terms.

$ {H_k} = \dfrac{k}{{\dfrac{1}{{{a_1}}} + \dfrac{1}{{{a_2}}} + \dfrac{1}{{{a_3}}} + .....\dfrac{1}{{{a_k}}}}}..........(6) $

$ {H_k} = \dfrac{{{a_1}k}}{{1 + \dfrac{1}{r} + ....... + \dfrac{1}{{{r^{k - 1}}}}}}..........(7) $

$ {H_k} = \dfrac{{{a_1}k(r - 1) \times {r^{k - 1}}}}{{{r^{k - 1}}}}..........(8) $

From Equations $ 2,5,8 $ ,we get the following relation between $ {G_k},{H_k},{A_k} $

$ {G_k} = {({A_k}{H_k})^{\dfrac{1}{2}}} $

Considering there are infinite number of terms , equation will transform as follows

$ {\prod\limits_{k = 1}^n G _k} = \prod\limits_{k = 1}^n {{{({A_k}{H_k})}^{\dfrac{1}{2}}}} ................(9) $

Thus expanding RHS of equation $ 9 $ we get following relation

\[{\prod\limits_{k = 1}^n G _k} = {({A_1}{A_2}.......{A_n} \times {H_1}{H_2}........{H_n})^{\dfrac{1}{{2n}}}}\]

Thus the relation of geometric mean in terms of arithmetic mean and Harmonic mean is

\[{\prod\limits_{k = 1}^n G _k} = {({A_1}{A_2}.......{A_n} \times {H_1}{H_2}........{H_n})^{\dfrac{1}{{2n}}}}\]

**So, the correct answer is “\[{\prod\limits_{k = 1}^n G _k} = {({A_1}{A_2}.......{A_n} \times {H_1}{H_2}........{H_n})^{\dfrac{1}{{2n}}}}\]**

”.

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**Note**: Though this sum looks extremely complicated and difficult to solve, it is easy if the approach is correct. Students are advised to memorize the formula for Arithmetic Mean , Geometric Mean , Harmonic mean for sum upto $ n $ terms. The sum from this chapter should be picked up last if it is of similar type. This is because if the approach is wrong for the sum , it will lead to complete waste of time. This sum is important for Students who are good with application and like to take up challenging numericals.

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