What is the harmonic mean of the reciprocals | vedclass

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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?

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