Find the Geometric Mean (G.M.) of the sequenc | vedclass

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Question

(D) The sequence is $1, 2, 2^2, \dots, 2^n$. The total number of terms is $n+1$.
The Geometric Mean $(G.M.)$ is given by the $(n+1)^{th}$ root of the

Vedclass · Accepted Answer

$2^{n/2}$

Vedclass · Answer

(D) The sequence is $1, 2, 2^2, \dots, 2^n$. The total number of terms is $n+1$.
The Geometric Mean $(G.M.)$ is given by the $(n+1)^{th}$ root of the product of all terms:
$G.M. = (1 	imes 2 	imes 2^2 	imes \dots 	imes 2^n)^{\frac{1}{n+1}}$
The product of the terms is $2^{(0+1+2+\dots+n)}$. Using the sum of the first $n$ natural numbers formula $\frac{n(n+1)}{2}$,we get:
$G.M. = (2^{\frac{n(n+1)}{2}})^{\frac{1}{n+1}}$
Simplifying the exponent:
$G.M. = 2^{\frac{n(n+1)}{2(n+1)}} = 2^{n/2}$

Find the Geometric Mean $(G.M.)$ of the sequence $1, 2, 2^2, \dots, 2^n$.

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