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

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

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 .....

Similar Questions

If $x > 1, y > 1, z > 1$ are in geometric progression,then in which progression are $\frac{1}{1 + \ln x}, \frac{1}{1 + \ln y}, \frac{1}{1 + \ln z}$?

The sum of the first ten terms of a geometric progression is $S_1$ and the sum of the next ten terms ($11^{th}$ to $20^{th}$) is $S_2$. What is the common ratio?

If $A = 1 + r^z + r^{2z} + r^{3z} + \dots \infty$,then the value of $r$ is:

Three numbers are selected from the set $\{3^1, 3^2, 3^3, \dots, 3^{20}\}$. Find the number of ways the selected numbers can form an increasing $G.P.$

If ${x_1}, {x_2}, {x_3}$ as well as ${y_1}, {y_2}, {y_3}$ are in $G$.$P$. with the same common ratio,then the points $({x_1}, {y_1}), ({x_2}, {y_2})$ and $({x_3}, {y_3})$:

Vedclass Test Series

Exam Paper Generator

Online Exam Module