The harmonic mean of 4, 8 and 16 is (Rounded off to four decimal places)

The correct option is 6.8571.

To find the harmonic mean of a set of numbers, we use the formula for the harmonic mean (HM) of n numbers:

$$\text{HM}=\frac{n}{\frac{1}{x_1}+\frac{1}{x_2}+\cdots +\frac{1}{x_n}}$$

For the given numbers 4, 8, and 16, the total number of values is n = 3. Substituting these values into the formula, we get:

$$\text{HM}=\frac{3}{\frac{1}{4}+\frac{1}{8}+\frac{1}{16}}$$

First, we find a common denominator for the fractions in the denominator, which is 16:

$$\frac{1}{4}=\frac{4}{16}$$

$$\frac{1}{8}=\frac{2}{16}$$

Now, add the fractions together:

$$\frac{4}{16}+\frac{2}{16}+\frac{1}{16}=\frac{7}{16}$$

Substitute this sum back into the harmonic mean formula:

$$\text{HM}=\frac{3}{\left(\frac{7}{16}\right)}=3\times \frac{16}{7}=\frac{48}{7}$$

Dividing 48 by 7 gives:

$$\frac{48}{7}\approx \mathrm{6.8571428...}$$

Rounding this to four decimal places, we get 6.8571. Therefore, the harmonic mean of 4, 8, and 16 is 6.8571.