Sequences And Series Ques 24
- Let $a_1, a_{2 r}$. be positive real numbers in geometric progression. For each $n$, if $A_n, G_n, H_n$ are respectively, the arithmetic mean, geometric mean and harmonic mean of $a_1, a_2, \ldots, a_n$. Then, find an expression for the geometric mean of $G_1, G_2, \ldots, G_n$ in terms of $A_1, A_2, \ldots, A_n, H_1, H_2, \ldots, H_n$.
(2001, 5M)
Show Answer
Solution: Let $G_m$ be the geometric mean of $G_1, G_2, \ldots, G_n$.
$ \begin{aligned} \Rightarrow \quad G_m= & \left(G_1 \cdot G_2 \ldots G_n\right)^{1 / n} \\ = & {\left[\left(a_1\right) \cdot\left(a_1 \cdot a_1 r\right)^{1 / 2} \cdot\left(a_1 \cdot a_1 r \cdot a_1 r^2\right)^{1 / 3}\right.} \\ & \left.\quad \cdots\left(a_1 \cdot a_1 r \cdot a_1 r^2 \ldots a_1 r^{n-1}\right)^{1 / n}\right]^{1 / n} \end{aligned} $
where, $r$ is the common ratio of GP $a_1, a_2, \ldots, a_n$.
$=\left[\left(a_1 \cdot a_1 \ldots n \text { times }\right)\left(r^{1 / 2} \cdot r^{3 / 3} \cdot r^{6 / 4} \ldots r^{\frac{(n-1) n}{2 n}}\right)\right]^{1 / n} $
$=\left[a_1^n \cdot r^{\frac{1}{2}+1+\frac{3}{2}+\cdots+\frac{n-1}{2}}\right]^{1 / n} $
$=a_1\left[r^{\frac{1}{2}}\left[\frac{(n-1) n}{2}\right]\right]^{1 / n}=a_1\left[r^{\frac{n-1}{4}}\right] \quad \ldots \text { (i) }$
Now, $\quad A_n=\frac{a_1+a_2+\ldots+a_n}{n}=\frac{a_1\left(1-r^n\right)}{n(1-r)}$
and $ \quad H_n=\frac{n}{\left(\frac{1}{a_1}+\frac{1}{a_2}+\ldots+\frac{1}{a_n}\right)}$
$=\frac{n}{\frac{1}{a_1}\left(1+\frac{1}{r}+\ldots+\frac{1}{r^{n-1}}\right)}$
$=\frac{a_1 n(1-r) r^{n-1}}{1-r^n}$
$ \begin{aligned} \therefore \quad A_n \cdot H_n & =\frac{a_1\left(1-r^n\right)}{n(1-r)} \times \frac{a_1 n(1-r) r^{n-1}}{\left(1-r^n\right)}=a_1^2 r^{n-1} \\ \Rightarrow \quad \prod_{k=1}^n A_k H_k & =\prod_{k=1}^n\left(a_1^2 r^{n-1}\right) \\ & =\left(a_1^2 \cdot a_1^2 \cdot a_1^2 \ldots n \text { times }\right) \times r^0 \cdot r^1 \cdot r^2 \ldots r^{n-1} \\ & =a_1^{2 n} \cdot r^{1+2+\ldots+(n-1)} \\ & =a_1^{2 n} r \frac{n(n-1)}{2}=\left[a_1 r^{\frac{n-1}{4}}\right]^{2 n} \end{aligned} $
$ =\left[G_m\right]^{2 n} \quad $ [from Eq. (i)]
$G_m =\left[\prod_{k=1}^n A_k H_k\right]^{1 / 2 n}$
$ \Rightarrow \quad G_m=\left(A_1 A_2 \ldots A_n H_1 H_2 \ldots H_n\right)^{1 / 2 n} $
BETA
