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# Let ${a_1},{a_2}.........$ be positive real numbers in geometric progression. For each $n$ , let ${A_n},{G_n},{H_n}$ be respectively, the arithmetic mean, geometric mean & harmonic mean of ${a_1},{a_2}.........{a_n}$ . Find the expression for the geometric mean of ${G_1},{G_2},{G_3}........{G_n}$ in terms of ${A_1},{A_2},{A_3}.........{A_n},{H_1},{H_2},{H_3}..........{H_n}$

Last updated date: 13th Sep 2024
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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.

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

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.