The sequence `{x_(1),x_(2),…x_(50)}` has the property that for each `k`, `x_(k)` is `k` less than the sum of other `49` numbers. The value of `96x_(20)` is

Options

A. `300`
B. `315`
C. `1024`
D. `0`

Explanation

To solve the problem, we will follow these steps:

1. **Understanding the property of the sequence**: We have a sequence \( \{x_1, x_2, \ldots, x_{50}\} \) such that for each \( k \), \( x_k \) is \( k \) less than the sum of the other 49 numbers. This can be expressed mathematically as:
\[ x_k = S - x_k - k \]
where \( S \) is the total sum of the sequence, \( S = x_1 + x_2 + \ldots + x_{50} \).

2. **Rearranging the equation**: From the equation above, we can rearrange it to:
\[ 2x_k + k = S \]
This implies:
\[ 2x_k = S - k \]
Therefore, we can express \( x_k \) as:
\[ x_k = \frac{S - k}{2} \]

3. **Summing over all \( k \)**: Now, we will sum this equation for \( k \) from 1 to 50:
\[ \sum_{k=1}^{50} x_k = \sum_{k=1}^{50} \frac{S - k}{2} \]
This simplifies to:
\[ \sum_{k=1}^{50} x_k = \frac{1}{2} \left( 50S - \sum_{k=1}^{50} k \right) \]
The sum of the first 50 natural numbers is given by:
\[ \sum_{k=1}^{50} k = \frac{50 \times 51}{2} = 1275 \]
Thus, we have:
\[ \sum_{k=1}^{50} x_k = \frac{1}{2} \left( 50S - 1275 \right) \]

4. **Setting the sum equal to \( S \)**: Since \( \sum_{k=1}^{50} x_k = S \), we can set the equations equal:
\[ S = \frac{1}{2} \left( 50S - 1275 \right) \]
Multiplying both sides by 2 gives:
\[ 2S = 50S - 1275 \]
Rearranging this gives:
\[ 48S = 1275 \]
Therefore, we find:
\[ S = \frac{1275}{48} \]

5. **Finding \( x_{20} \)**: Now we can substitute \( S \) back to find \( x_{20} \):
\[ x_{20} = \frac{S - 20}{2} = \frac{\frac{1275}{48} - 20}{2} \]
To simplify \( \frac{1275}{48} - 20 \):
\[ 20 = \frac{960}{48} \quad \text{(since } 20 \times 48 = 960\text{)} \]
Thus:
\[ x_{20} = \frac{\frac{1275 - 960}{48}}{2} = \frac{\frac{315}{48}}{2} = \frac{315}{96} \]

6. **Calculating \( 96x_{20} \)**: Finally, we calculate \( 96x_{20} \):
\[ 96x_{20} = 96 \times \frac{315}{96} = 315 \]

Thus, the value of \( 96x_{20} \) is \( \boxed{315} \).