Sum of the reciprocals of all the 100 harmonic means if these are inserted between 1 and 1/100:

To find the sum of the reciprocals of all the 100 harmonic means inserted between 1 and 1/100, we can start by understanding what a harmonic mean is.

Harmonic Mean:
The harmonic mean is a type of average that is calculated by dividing the number of values by the sum of their reciprocals. For a series of numbers, the harmonic mean is the reciprocal of the arithmetic mean of the reciprocals of the numbers.

Inserting Harmonic Means:
When we insert harmonic means between two numbers, we are essentially dividing the interval between the numbers into equal parts.

In this case, we are inserting 100 harmonic means between 1 and 1/100. Let's denote the harmonic means as H1, H2, H3, ..., H100.

Finding the Harmonic Means:
To find the harmonic means, we can use the formula:

Hn = 2/(1/A + 1/B)

Where A and B are the two numbers between which we are inserting the harmonic mean.

In this case, A = 1 and B = 1/100. Substituting the values into the formula, we get:

Hn = 2/(1/1 + 1/(1/100))
= 2/(1 + 100)
= 2/101

So, each harmonic mean is equal to 2/101.

Calculating the Sum:
To calculate the sum of the reciprocals of all the 100 harmonic means, we need to find the sum of the reciprocals of H1, H2, H3, ..., H100.

Since each harmonic mean is 2/101, we can write the sum as:

Sum = 1/H1 + 1/H2 + 1/H3 + ... + 1/H100
= (101/2) + (101/2) + (101/2) + ... + (101/2)
= 100 * (101/2)
= 5050

Therefore, the sum of the reciprocals of all the 100 harmonic means inserted between 1 and 1/100 is 5050.