The harmonic mean of the numbers 1 , (1/2) , (1/3) , (1/4),...., (1/n) is

Question

The harmonic mean of the numbers 1 , (1/2) , (1/3) , (1/4),...., (1/n) is (A) (n + 1/2)  (B) (2/n + 1)  (C) (n/1 +(1)2 + (1)3 + (1/4)+....+ (1/n) (D)  (1 + (1/2) + (1/3) + (1/4)+....+ (1/n)/n)

Tardigrade Admin · Accepted Answer

Harmonic mean of x1, x2, ... , xn (none of which is 0) = (n/∑((1)xi)) ∴ Required harmonic mean = (n/ 1+2+3+..+n) = (2n/n(n+1)) = (2/n+1)

Q. The harmonic mean of the numbers $1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, ...., \frac{1}{n}$ is

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

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

$\frac{n}{1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + .... + \frac{1}{n}}$

$\frac{1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + .... + \frac{1}{n}}{n}$

Harmonic mean of $x_{1}, x_{2}, ..., x_{n}$ (none of which is 0)
$= \frac{n}{\sum ( \frac{1}{x _{i}} )}$
$∴$ Required harmonic mean
$= \frac{n}{1 + 2 + 3 + .. + n} = \frac{2 n}{n ( n + 1 )} = \frac{2}{n + 1}$

Q. The harmonic mean of the numbers 1,21​,31​,41​,....,n1​ is

2n+1​

n+12​

1+21​+31​+41​+....+n1​n​

n1+21​+31​+41​+....+n1​​