The geometric mean of the first n terms of th | vedclass

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Question

(C) The sequence is a geometric progression: $a, ar, ar^2, \dots, ar^{n-1}$.
The geometric mean $(GM)$ of $n$ terms $x_1, x_2, \dots, x_n$ is given b

Vedclass · Accepted Answer

$ar^{(n-1)/2}$

Vedclass · Answer

(C) The sequence is a geometric progression: $a, ar, ar^2, \dots, ar^{n-1}$.
The geometric mean $(GM)$ of $n$ terms $x_1, x_2, \dots, x_n$ is given by $(x_1 	imes x_2 	imes \dots 	imes x_n)^{1/n}$.
Here,$GM = (a 	imes ar 	imes ar^2 	imes \dots 	imes ar^{n-1})^{1/n}$.
$GM = (a^n 	imes r^{(0+1+2+\dots+(n-1))})^{1/n}$.
The sum of the first $(n-1)$ integers is $\frac{(n-1)n}{2}$.
$GM = (a^n 	imes r^{n(n-1)/2})^{1/n}$.
$GM = a 	imes r^{(n-1)/2} = ar^{(n-1)/2}$.

The geometric mean of the first $n$ terms of the sequence $a, ar, ar^2, \dots$ is:

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