Question

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 $A_1,A_2,A_3.........A_n,H_1,H_2,H_3..........H_n$

Answer 1

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.

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
 $G_k=(a_{1}r{)}^{{\displaystyle \frac{k-1}{2}}}..............(2)$
Following is the formula for Arithmetic progression upto $k$ terms.
 $A_k={\displaystyle \frac{a_1+a_2+......a_k}{k}}..........(3)$
 $A_k={\displaystyle \frac{a_1(1+r+.......r^{k-1})}{k}}..........(4)$
 $A_k={\displaystyle \frac{a_1(r^k-1)}{(r-1)k}}..........(5)$
Noting down the formula for Harmonic Progression upto $k$ terms.
 $H_k={\displaystyle \frac{k}{{\displaystyle \frac{1}{a_1}}+{\displaystyle \frac{1}{a_2}}+{\displaystyle \frac{1}{a_3}}+.....{\displaystyle \frac{1}{a_k}}}}..........(6)$
 $H_k={\displaystyle \frac{a_{1}k}{1+{\displaystyle \frac{1}{r}}+.......+{\displaystyle \frac{1}{r^{k-1}}}}}..........(7)$
 $H_k={\displaystyle \frac{a_{1}k(r-1)\times r^{k-1}}{r^{k-1}}}..........(8)$
From Equations $2,5,8$ ,we get the following relation between $G_k,H_k,A_k$
 $G_k=(A_k{H}_k{)}^{{\displaystyle \frac{1}{2}}}$
Considering there are infinite number of terms , equation will transform as follows
 $\prod _{k=1}^n{G}_k=\prod _{k=1}^n{(A_k{H}_k)}^{{\displaystyle \frac{1}{2}}}................(9)$
Thus expanding RHS of equation $9$ we get following relation
$$\prod _{k=1}^n{G}_k=(A_1{A}_2.......A_n\times H_1{H}_2........H_n{)}^{{\displaystyle \frac{1}{2n}}}$$
Thus the relation of geometric mean in terms of arithmetic mean and Harmonic mean is
$$\prod _{k=1}^n{G}_k=(A_1{A}_2.......A_n\times H_1{H}_2........H_n{)}^{{\displaystyle \frac{1}{2n}}}$$
So, the correct answer is “$$\prod _{k=1}^n{G}_k=(A_1{A}_2.......A_n\times H_1{H}_2........H_n{)}^{{\displaystyle \frac{1}{2n}}}$$
”.

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.