`sum_(k=0)^(5)(-1)^(k)2k`

Options

A. `-10`
B. `-6`
C. 0
D. 6

Explanation

To solve the summation \( \sum_{k=0}^{5} (-1)^{k} 2k \), we will evaluate each term of the series step by step.

### Step-by-Step Solution:

1. **Write down the summation**:
\[ \sum_{k=0}^{5} (-1)^{k} 2k = (-1)^{0} 2 \cdot 0 + (-1)^{1} 2 \cdot 1 + (-1)^{2} 2 \cdot 2 + (-1)^{3} 2 \cdot 3 + (-1)^{4} 2 \cdot 4 + (-1)^{5} 2 \cdot 5 \]

2. **Calculate each term**:
- For \( k = 0 \):
\[ (-1)^{0} \cdot 2 \cdot 0 = 1 \cdot 0 = 0 \]
- For \( k = 1 \):
\[ (-1)^{1} \cdot 2 \cdot 1 = -1 \cdot 2 = -2 \]
- For \( k = 2 \):
\[ (-1)^{2} \cdot 2 \cdot 2 = 1 \cdot 4 = 4 \]
- For \( k = 3 \):
\[ (-1)^{3} \cdot 2 \cdot 3 = -1 \cdot 6 = -6 \]
- For \( k = 4 \):
\[ (-1)^{4} \cdot 2 \cdot 4 = 1 \cdot 8 = 8 \]
- For \( k = 5 \):
\[ (-1)^{5} \cdot 2 \cdot 5 = -1 \cdot 10 = -10 \]

3. **Combine the results**:
Now we sum all the calculated terms:
\[ 0 + (-2) + 4 + (-6) + 8 + (-10) \]

4. **Group the terms**:
- Positive terms: \( 4 + 8 = 12 \)
- Negative terms: \( -2 - 6 - 10 = -18 \)

5. **Final calculation**:
\[ 12 + (-18) = 12 - 18 = -6 \]

Thus, the final answer is:
\[ \sum_{k=0}^{5} (-1)^{k} 2k = -6 \]

### Conclusion:
The correct option is **Option 2: -6**.