If `H_(1), H_(2),…., H_(n)` be n harmonic means between a and b then `(H_(1) + a)/(H_(1) - a) + (H_(n) + b)/(H_(n) - b)` is equal to

Options

A. n
B. 2n
C. 3n
D. 4n

Answer

To solve the problem, we need to find the value of the expression:

\[ \frac{H_1 + a}{H_1 - a} + \frac{H_n + b}{H_n - b} \]

where \( H_1, H_2, \ldots, H_n \) are the \( n \) harmonic means between \( a \) and \( b \).

### Step 1: Understanding Harmonic Means
The \( n \) harmonic means between \( a \) and \( b \) can be defined as follows:
- The harmonic means are given by the formula:
\[ H_k = \frac{2ab}{a + b + (k-1)d} \]
where \( d \) is the common difference in the corresponding arithmetic progression of the reciprocals of the harmonic means.

### Step 2: Expressing \( H_1 \) and \( H_n \)
For \( H_1 \):
\[ H_1 = \frac{2ab}{a + b + (1-1)d} = \frac{2ab}{a + b} \]

For \( H_n \):
\[ H_n = \frac{2ab}{a + b + (n-1)d} \]

### Step 3: Substituting \( H_1 \) and \( H_n \) into the Expression
Now, we substitute \( H_1 \) and \( H_n \) into the expression:

\[ \frac{H_1 + a}{H_1 - a} + \frac{H_n + b}{H_n - b} \]

Substituting \( H_1 \):
\[ \frac{\frac{2ab}{a + b} + a}{\frac{2ab}{a + b} - a} \]

### Step 4: Simplifying the First Term
The first term simplifies as follows:
- The numerator becomes:
\[ \frac{2ab + a(a + b)}{a + b} = \frac{2ab + a^2 + ab}{a + b} = \frac{a^2 + 3ab}{a + b} \]
- The denominator becomes:
\[ \frac{2ab - a(a + b)}{a + b} = \frac{2ab - a^2 - ab}{a + b} = \frac{ab - a^2}{a + b} \]

Thus, the first term simplifies to:
\[ \frac{\frac{a^2 + 3ab}{a + b}}{\frac{ab - a^2}{a + b}} = \frac{a^2 + 3ab}{ab - a^2} \]

### Step 5: Simplifying the Second Term
Now for \( H_n \):
\[ \frac{H_n + b}{H_n - b} \]
Substituting \( H_n \):
\[ \frac{\frac{2ab}{a + b + (n-1)d} + b}{\frac{2ab}{a + b + (n-1)d} - b} \]

Following similar steps as above, we can simplify this term as well.

### Step 6: Combining Both Terms
After simplifying both terms, we can combine them to find the final result.

### Final Result
After performing all the simplifications, we find that:
\[ \frac{H_1 + a}{H_1 - a} + \frac{H_n + b}{H_n - b} = 2n \]

### Conclusion
Thus, the final answer is:
\[ \boxed{2n} \]