What is the harmonic mean of the reciprocals | vedclass

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

Question

(C) The first $n$ natural numbers are $1, 2, 3, \dots, n$. Their reciprocals are $1, \frac{1}{2}, \frac{1}{3}, \dots, \frac{1}{n}$. The harmonic mean

Vedclass · Accepted Answer

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

Vedclass · Answer

(C) The first $n$ natural numbers are $1, 2, 3, \dots, n$. Their reciprocals are $1, \frac{1}{2}, \frac{1}{3}, \dots, \frac{1}{n}$. The harmonic mean $(HM)$ of a set of $n$ numbers $x_1, x_2, \dots, x_n$ is given by $HM = \frac{n}{\sum_{i=1}^{n} \frac{1}{x_i}}$. Here,$x_i = \frac{1}{i}$ for $i = 1, 2, \dots, n$. So,$\frac{1}{x_i} = i$. Therefore,$HM = \frac{n}{\sum_{i=1}^{n} i} = \frac{n}{\frac{n(n+1)}{2}} = \frac{2n}{n(n+1)} = \frac{2}{n+1}$.

What is the harmonic mean of the reciprocals of the first $n$ natural numbers?

Similar Questions

If $\cos (\theta-\alpha), \cos \theta$ and $\cos (\theta+\alpha)$ are in harmonic progression,then $2 \tan ^2 \theta=$

If $\frac{x + y}{2}, y, \frac{y + z}{2}$ are in $H.P.$,then $x, y, z$ are in

The terms $\log _3 2, \log _6 2, \log _{12} 2$ are in:

If $a_1, a_2, a_3, \dots, a_n$ form a harmonic progression,find the value of $a_1a_2 + a_2a_3 + \dots + a_{n-1}a_n$.

If the $7^{th}$ term of a harmonic progression is $8$ and the $8^{th}$ term is $7$,then its $15^{th}$ term is

Vedclass Test Series

Exam Paper Generator

Online Exam Module