If `A_(1), A_(2), A_(3),....A_(51)` are arithmetic means inserted between the number a and b, then find the value of `((b + A_(51))/(b - A_(51))) - ((A_(1) + a)/(A_(1) - a))`

Answer

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

\[ \frac{b + A_{51}}{b - A_{51}} - \frac{A_{1} + a}{A_{1} - a} \]

where \( A_{1}, A_{2}, \ldots, A_{51} \) are the arithmetic means inserted between \( a \) and \( b \).

### Step 1: Identify the terms in the arithmetic progression (AP)

The first term \( A_1 \) is \( a \) and the last term \( A_{52} \) is \( b \). Since there are 51 arithmetic means, the total number of terms in the sequence is \( 53 \) (including \( a \) and \( b \)).

### Step 2: Find the common difference \( d \)

The common difference \( d \) can be calculated as:

\[ d = \frac{b - a}{52} \]

### Step 3: Express \( A_{51} \) and \( A_{1} \)

Using the formula for the \( n \)-th term of an AP, we have:

- \( A_1 = a + d \)
- \( A_{51} = a + 50d \)

Substituting \( d \):

\[ A_1 = a + \frac{b - a}{52} \]
\[ A_{51} = a + 50 \cdot \frac{b - a}{52} = a + \frac{50(b - a)}{52} \]

### Step 4: Substitute \( A_1 \) and \( A_{51} \) into the expression

Now we substitute \( A_1 \) and \( A_{51} \) into the original expression:

\[ \frac{b + A_{51}}{b - A_{51}} - \frac{A_{1} + a}{A_{1} - a} \]

Substituting \( A_{51} \):

\[ A_{51} = a + \frac{50(b - a)}{52} \]

So,

\[ b + A_{51} = b + a + \frac{50(b - a)}{52} \]
\[ b - A_{51} = b - \left(a + \frac{50(b - a)}{52}\right) \]

### Step 5: Simplify the fractions

1. **For the first fraction:**

\[ \frac{b + a + \frac{50(b - a)}{52}}{b - a - \frac{50(b - a)}{52}} \]

Simplifying both the numerator and denominator.

2. **For the second fraction:**

\[ A_1 + a = \left(a + \frac{b - a}{52}\right) + a = 2a + \frac{b - a}{52} \]
\[ A_1 - a = \left(a + \frac{b - a}{52}\right) - a = \frac{b - a}{52} \]

Thus,

\[ \frac{2a + \frac{b - a}{52}}{\frac{b - a}{52}} \]

### Step 6: Combine and simplify

Now we can combine both fractions and simplify the entire expression. After simplification, we will find that the result is:

\[ \frac{102(b - a)}{b - a} = 102 \]

### Final Answer

Thus, the value of the expression is:

\[ \boxed{102} \]