The geometric mean of 7, 7^2, 7^3, dots, 7^n | vedclass

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Question

(A) The geometric mean of $n$ terms $a_1, a_2, \dots, a_n$ is given by $(a_1 	imes a_2 	imes \dots 	imes a_n)^{\frac{1}{n}}$.For the sequence $7, 7

Vedclass · Accepted Answer

$7^{\frac{n+1}{2}}$

Vedclass · Answer

(A) The geometric mean of $n$ terms $a_1, a_2, \dots, a_n$ is given by $(a_1 	imes a_2 	imes \dots 	imes a_n)^{\frac{1}{n}}$.For the sequence $7, 7^2, 7^3, \dots, 7^n$,the geometric mean is $(7^1 	imes 7^2 	imes 7^3 	imes \dots 	imes 7^n)^{\frac{1}{n}}$.Using the property of exponents,this becomes $(7^{1+2+3+\dots+n})^{\frac{1}{n}}$.The sum of the first $n$ natural numbers is $\frac{n(n+1)}{2}$.Thus,the expression becomes $(7^{\frac{n(n+1)}{2}})^{\frac{1}{n}}$.Simplifying the exponents,we get $7^{\frac{n(n+1)}{2n}} = 7^{\frac{n+1}{2}}$.

The geometric mean of $7, 7^2, 7^3, \dots, 7^n$ is .....

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