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Q.
If a variate takes values $ a, ar, ar^2, ..., ar^{n-1} $ then which of the following relations between means hold ?
AMUAMU 2010Sequences and Series
Solution:
Given series $a, ar, ar^{2}, \dots, ar^{n-1}$
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
$G=\sqrt{a\cdot ar}$
$G^{2}=a^{2}\cdot r \ldots\left(i\right)$
Now, $AH=\frac{a\left(1+r\right)}{2}\cdot\frac{2ar}{\left(1+r\right)}=a^{2}r \ldots\left(ii\right)$
From Eqs. $\left(i\right)$ and $\left(ii\right)$, $G^{2}=A\cdot H$