The harmonic mean of 4,8,16 is

Options

A. 6.4
B. 6.7
C. 6.85
D. 7.8

Explanation

To find the harmonic mean of the numbers 4, 8, and 16, we will follow these steps:

### Step 1: Write down the formula for the harmonic mean (HM).
The formula for the harmonic mean of \( n \) numbers \( x_1, x_2, \ldots, x_n \) is given by:

\[ HM = \frac{n}{\frac{1}{x_1} + \frac{1}{x_2} + \frac{1}{x_3} + \ldots + \frac{1}{x_n}} \]

### Step 2: Identify the values of \( n \) and the numbers.
In this case, we have three numbers: 4, 8, and 16. Therefore, \( n = 3 \) and the numbers are \( x_1 = 4 \), \( x_2 = 8 \), and \( x_3 = 16 \).

### Step 3: Calculate the sum of the reciprocals of the numbers.
We need to calculate:

\[ \frac{1}{x_1} + \frac{1}{x_2} + \frac{1}{x_3} = \frac{1}{4} + \frac{1}{8} + \frac{1}{16} \]

### Step 4: Find a common denominator to add the fractions.
The least common multiple (LCM) of 4, 8, and 16 is 16. We will convert each fraction to have a denominator of 16:

\[ \frac{1}{4} = \frac{4}{16}, \quad \frac{1}{8} = \frac{2}{16}, \quad \frac{1}{16} = \frac{1}{16} \]

Now, we can add them:

\[ \frac{4}{16} + \frac{2}{16} + \frac{1}{16} = \frac{4 + 2 + 1}{16} = \frac{7}{16} \]

### Step 5: Substitute back into the harmonic mean formula.
Now we can substitute back into the harmonic mean formula:

\[ HM = \frac{3}{\frac{7}{16}} = 3 \times \frac{16}{7} = \frac{48}{7} \]

### Step 6: Calculate the final value.
Now, we can convert \( \frac{48}{7} \) to a decimal:

\[ \frac{48}{7} \approx 6.8571 \]

### Final Answer:
Thus, the harmonic mean of 4, 8, and 16 is approximately \( 6.8571 \).

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