Suppose p is the first of `n(ngt1)` arithmetic means between two positive numbers a and b and q the first of n harmonic means between the same two numbers.
The value of q is

Options

A. `((n-1)ab)/(nb+a)`
B. `((n+1)ab)/(nb+a)`
C. `((n-1)ab)/(na+b)`
D. `((n-1)ab)/(na+b)`

Explanation

To find the value of \( q \), which is the first of \( n \) harmonic means between two positive numbers \( a \) and \( b \), we will follow these steps:

### Step 1: Understand the Arithmetic Means
Given two positive numbers \( a \) and \( b \), if we insert \( n \) arithmetic means between them, the sequence becomes:
\[ a, A_1, A_2, \ldots, A_n, b \]
Here, \( A_1 \) is the first arithmetic mean, and it can be calculated using the formula for the \( k \)-th term of an arithmetic progression (AP):
\[ A_k = a + (k-1)d \]
where \( d \) is the common difference.

### Step 2: Calculate the Common Difference
The total number of terms in this AP is \( n + 2 \) (including \( a \) and \( b \)). The common difference \( d \) can be calculated as:
\[ d = \frac{b - a}{n + 1} \]

### Step 3: Find the First Arithmetic Mean \( p \)
The first arithmetic mean \( p \) (which is \( A_1 \)) is given by:
\[ p = A_1 = a + d = a + \frac{b - a}{n + 1} \]
Now, simplifying this:
\[ p = a + \frac{b - a}{n + 1} = \frac{a(n + 1) + (b - a)}{n + 1} = \frac{an + a + b - a}{n + 1} = \frac{an + b}{n + 1} \]

### Step 4: Understand the Harmonic Means
The harmonic means between \( a \) and \( b \) can be represented as:
\[ a, H_1, H_2, \ldots, H_n, b \]
The harmonic means are the reciprocals of an arithmetic progression formed by the reciprocals of \( a \) and \( b \):
\[ \frac{1}{a}, \frac{1}{H_1}, \frac{1}{H_2}, \ldots, \frac{1}{H_n}, \frac{1}{b} \]

### Step 5: Calculate the Common Difference for Harmonic Means
The common difference \( D \) for the reciprocals is:
\[ D = \frac{\frac{1}{b} - \frac{1}{a}}{n + 1} = \frac{a - b}{ab(n + 1)} \]

### Step 6: Find the First Harmonic Mean \( q \)
The first harmonic mean \( q \) (which is \( H_1 \)) can be calculated as:
\[ \frac{1}{H_1} = \frac{1}{a} + D \]
Substituting \( D \):
\[ \frac{1}{H_1} = \frac{1}{a} + \frac{a - b}{ab(n + 1)} \]
Now, simplifying this:
\[ \frac{1}{H_1} = \frac{b + (a - b)(n + 1)}{ab(n + 1)} = \frac{(n + 1)a + b - b}{ab(n + 1)} = \frac{(n + 1)a}{ab(n + 1)} \]
Thus, we find:
\[ H_1 = \frac{ab(n + 1)}{a + nb} \]

### Conclusion
The value of \( q \) (the first harmonic mean) is:
\[ q = \frac{ab(n + 1)}{a + nb} \]