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 A.M, G.M and H.M of $a_1,a_2,a_n$. Find an expression for the G.M of $G_1,G_2,...G_n$ in terms of $A_1,A_2,...A_n$ and $H_1,H_2,...H_n$

• A. $G_n=(A_2{A}_4.....A_n{H}_1{H}_2....H_{2n}{)}^{1/2n}$
• B. $G_n=(A_1{A}_2.....A_n{H}_2{H}_1....H_{2n}{)}^{1/2n}$
• C. $G_n=(A_1{A}_2.....A_n{H}_1{H}_2....H_n{)}^{1/2n}$
• D. None of these

Answer 1

The correct option is C $G_n=(A_1{A}_2.....A_n{H}_1{H}_2....H_n{)}^{1/2n}$

Let $G_m$ be the geometric mean of $G_1,G_2,...,G_n.$

$\Rightarrow G_m={(G_1.G_2...G_n)}^{\frac{1}{n}}$

$={[\left(a_1\right).{(a_1.a_{1}r)}^{\frac{1}{2}}.{(a_1.a_{1}r.a_1{r}^2)}^{\frac{1}{3}}...{(a_1.a_{1}r.a_1{r}^2...a_1{r}^{n-1})}^{\frac{1}{n}}]}^{\frac{1}{n}}$

where $r$ is the common ratio of $GP{a}_1,a_2,...,a_n.$

$={\left[(a_1.a_1...ntimes)(r^{\frac{1}{2}}.r^{\frac{3}{3}}.r^{\frac{6}{4}}...r^{\frac{(n-1)n}{2n}})\right]}^{\frac{1}{n}}$

$={[a_1^n.r^{\frac{1}{2}+1+\frac{3}{2}+...+\frac{n-1}{2}}]}^{\frac{1}{n}}$

$=a_1{\left[r^{\frac{1}{2}\left[\frac{(n-1)n}{2}\right]}\right]}^{\frac{1}{n}}=a_1\left[r^{\frac{n-1}{4}}\right]$ ...(1)

Now, $A_n=\frac{a_1+a_2+...+a_n}{n}=\frac{a_1(1-r^n)}{n(1-r)}$

and $H_n=\frac{n}{(\frac{1}{a_1}+\frac{1}{a_2}+...+\frac{1}{a_n})}=\frac{n}{\frac{1}{a_1}(1+\frac{1}{r}+...+\frac{1}{r^{n-1}})}=\frac{a_{1}n(1-r)r^{n-1}}{1-r^n}$

$\therefore A_n.H_n=\frac{a_1(1-r^n)}{n(1-r)}\times \frac{a_{1}n(1-r)r^{n-1}}{(1-r^n)}=a_1^2{r}^{n-1}$

$\prod _{k=1}^n{A}_k{H}_k=\prod _{k=1}^n\left(a_1^2{r}^{n-1}\right)$

$=(a_1^2.a_1^2.a_1^2...ntimes)\times r^0.r^1.r^2...r^{n-1}$

$=a_1^{2n}.r^{1+2+...+(n-1)}=a_1^{2n}{r}^{\frac{n(n-1)}{2}}$

$={\left[a_1{r}^{\frac{n-1}{4}}\right]}^{2n}={\left[G_m\right]}^{2n}$ [from Eq.(1)]

$G_m={\left[\prod _{k=1}^n{A}_k{H}_k\right]}^{\frac{-1}{2n}}$

$\Rightarrow G_m={(A_1{A}_2...A_n{H}_1{H}_2...H_n)}^{\frac{1}{2n}}$