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

In a $H.P.$,the $p^{th}$ term is $q$ and the $q^{th}$ term is $p$. Then the $(pq)^{th}$ term is

Given $a + d > b + c$ where $a, b, c, d$ are real numbers,then

If $\frac{a^{n + 1} + b^{n + 1}}{a^n + b^n}$ is the harmonic mean between $a$ and $b$,then the value of $n$ is

The harmonic mean of $4, 8, 16$ is

Let $a, b, c, p, q$ and $r$ be positive real numbers such that $a, b$ and $c$ are in $GP$ and $a^{p} = b^{q} = c^{r}$. Then,

Vedclass Test Series

Exam Paper Generator

Online Exam Module