The H.M. of the numbers 1/5, 1/10, 1/15, 1/20, 1/25, 1/30, 1/35 is

Options

A. `1//20`
B. `1//16`
C. `1//15`
D. `1//13`

Answer

To find the Harmonic Mean (H.M.) of the numbers \( \frac{1}{5}, \frac{1}{10}, \frac{1}{15}, \frac{1}{20}, \frac{1}{25}, \frac{1}{30}, \frac{1}{35} \), we can follow these steps:

### Step 1: Identify the numbers
The given numbers are:
- \( A_1 = \frac{1}{5} \)
- \( A_2 = \frac{1}{10} \)
- \( A_3 = \frac{1}{15} \)
- \( A_4 = \frac{1}{20} \)
- \( A_5 = \frac{1}{25} \)
- \( A_6 = \frac{1}{30} \)
- \( A_7 = \frac{1}{35} \)

### Step 2: Count the total numbers
We have a total of \( N = 7 \) numbers.

### Step 3: Calculate the sum of the reciprocals
Next, we need to calculate the sum of the reciprocals of these numbers:
\[ \frac{1}{A_1} + \frac{1}{A_2} + \frac{1}{A_3} + \frac{1}{A_4} + \frac{1}{A_5} + \frac{1}{A_6} + \frac{1}{A_7} \]
Calculating each reciprocal:
- \( \frac{1}{A_1} = 5 \)
- \( \frac{1}{A_2} = 10 \)
- \( \frac{1}{A_3} = 15 \)
- \( \frac{1}{A_4} = 20 \)
- \( \frac{1}{A_5} = 25 \)
- \( \frac{1}{A_6} = 30 \)
- \( \frac{1}{A_7} = 35 \)

Now, summing these values:
\[ 5 + 10 + 15 + 20 + 25 + 30 + 35 = 5 + 10 = 15 \]
\[ 15 + 15 = 30 \]
\[ 30 + 20 = 50 \]
\[ 50 + 25 = 75 \]
\[ 75 + 30 = 105 \]
\[ 105 + 35 = 140 \]

So, the sum of the reciprocals is \( 140 \).

### Step 4: Apply the formula for Harmonic Mean
The formula for the Harmonic Mean is given by:
\[ H.M. = \frac{N}{\frac{1}{A_1} + \frac{1}{A_2} + \frac{1}{A_3} + \ldots + \frac{1}{A_N}} \]
Substituting the values we have:
\[ H.M. = \frac{7}{140} \]

### Step 5: Simplify the result
Now, simplify \( \frac{7}{140} \):
\[ H.M. = \frac{1}{20} \]

### Final Answer
Thus, the Harmonic Mean of the numbers \( \frac{1}{5}, \frac{1}{10}, \frac{1}{15}, \frac{1}{20}, \frac{1}{25}, \frac{1}{30}, \frac{1}{35} \) is \( \frac{1}{20} \).

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