If `(48)/(2.3)+(47)/(3.4)+(46)/(4.5)+ . . . +(2)/(48.49)+(1)/(49.50)`
`=(51)/(2)+k(1+(1)/(2)+(1)/(3)+ . . .+(1)/(50))`, then k equals

Options

A. 2
B. `-1`
C. `-(1)/(2)`
D. 1

Explanation

To solve the problem, we need to evaluate the sum on the left-hand side and compare it to the expression on the right-hand side to find the value of \( k \).

### Step-by-step Solution:

1. **Understanding the Series**:
The series we need to evaluate is:
\[ S = \frac{48}{2 \cdot 3} + \frac{47}{3 \cdot 4} + \frac{46}{4 \cdot 5} + \ldots + \frac{2}{48 \cdot 49} + \frac{1}{49 \cdot 50} \]

2. **Identifying the General Term**:
The general term of the series can be expressed as:
\[ t_r = \frac{50 - r}{r(r + 1)} \]
where \( r \) ranges from 2 to 49.

3. **Simplifying the General Term**:
We can separate the general term:
\[ t_r = \frac{50}{r(r + 1)} - \frac{r}{r(r + 1)} = \frac{50}{r(r + 1)} - \frac{1}{r + 1} \]

4. **Summing the Series**:
Now we can write the sum \( S \) as:
\[ S = \sum_{r=2}^{49} t_r = \sum_{r=2}^{49} \left( \frac{50}{r(r + 1)} - \frac{1}{r + 1} \right) \]

5. **Calculating the First Part**:
The first part can be simplified using the method of partial fractions:
\[ \sum_{r=2}^{49} \frac{50}{r(r + 1)} = 50 \left( \sum_{r=2}^{49} \left( \frac{1}{r} - \frac{1}{r + 1} \right) \right) \]
This is a telescoping series, which simplifies to:
\[ 50 \left( \frac{1}{2} - \frac{1}{50} \right) = 50 \left( \frac{25 - 1}{50} \right) = 24 \]

6. **Calculating the Second Part**:
The second part is:
\[ \sum_{r=2}^{49} \frac{1}{r + 1} = \sum_{s=3}^{50} \frac{1}{s} = H_{50} - H_{2} \]
where \( H_n \) is the \( n \)-th harmonic number.

7. **Combining the Results**:
Therefore, we have:
\[ S = 24 - (H_{50} - H_{2}) \]

8. **Setting Up the Equation**:
We know from the problem statement that:
\[ S = \frac{51}{2} + k \left( 1 + \frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{50} \right) \]
This can be rewritten as:
\[ S = \frac{51}{2} + k H_{50} \]

9. **Equating the Two Expressions**:
Now we equate the two expressions for \( S \):
\[ 24 - (H_{50} - H_{2}) = \frac{51}{2} + k H_{50} \]

10. **Solving for \( k \)**:
Rearranging gives:
\[ 24 + H_{2} - \frac{51}{2} = (k + 1) H_{50} \]
Simplifying further:
\[ \frac{48 - 51}{2} + H_{2} = (k + 1) H_{50} \]
\[ -\frac{3}{2} + H_{2} = (k + 1) H_{50} \]

11. **Finding \( k \)**:
Since \( H_{2} = 1 + \frac{1}{2} = \frac{3}{2} \):
\[ 0 = (k + 1) H_{50} \]
This implies \( k + 1 = 0 \) or \( k = -1 \).

Thus, the value of \( k \) is:
\[ \boxed{-1} \]