Question

If a variate takes values $a,ar,a{r}^2,...,a{r}^{n-1}$ then which of the following relations between means hold ?

• A. $A.H=G^2$
• B. $\frac{A+H}{2}=G$
• C. $A>G>H$
• D. $A=G=H$

Answer 1

Answer: (A)

Given series $a,ar,a{r}^2,\dots ,a{r}^{n-1}$
Then, arithmetic mean of first two terms
$A=\frac{a+ar}{2}$
$A=\frac{a\left(1+r\right)}{2}$
Harmonic mean of first two terms
$H=\frac{2\cdot a\cdot ar}{a+ar}$
$=\frac{2ar}{\left(1+r\right)}$
and geometric mean of first two terms
$G=\sqrt{a\cdot ar}$
$G^2=a^2\cdot r\dots \left(i\right)$
Now, $AH=\frac{a\left(1+r\right)}{2}\cdot \frac{2ar}{\left(1+r\right)}=a^{2}r\dots \left(ii\right)$
From Eqs. $\left(i\right)$ and $\left(ii\right)$, $G^2=A\cdot H$