Find the sum `S = sum _( k =1) ^( 2016) (-1) ^(( (k +1))/(2))k`

Answer

To solve the problem, we need to find the sum

\[ S = \sum_{k=1}^{2016} (-1)^{\frac{k+1}{2}} k. \]

### Step-by-Step Solution:

1. **Understanding the Summation**:
The expression \((-1)^{\frac{k+1}{2}}\) alternates between positive and negative values depending on whether \(k\) is odd or even. Specifically:
- For odd \(k\), \((-1)^{\frac{k+1}{2}} = 1\)
- For even \(k\), \((-1)^{\frac{k+1}{2}} = -1\)

2. **Separating the Odd and Even Terms**:
We can separate the sum into two parts: one for odd \(k\) and one for even \(k\).
- Odd \(k\): \(k = 1, 3, 5, \ldots, 2015\)
- Even \(k\): \(k = 2, 4, 6, \ldots, 2016\)

Thus, we can rewrite \(S\) as:
\[ S = \sum_{k \text{ odd}} k - \sum_{k \text{ even}} k \]

3. **Calculating the Odd Sum**:
The odd numbers from 1 to 2015 can be expressed as:
\[ 1, 3, 5, \ldots, 2015 \]
This forms an arithmetic series where:
- First term \(a = 1\)
- Last term \(l = 2015\)
- Common difference \(d = 2\)

The number of terms \(n\) can be found using:
\[ n = \frac{l - a}{d} + 1 = \frac{2015 - 1}{2} + 1 = 1008 \]

The sum of the first \(n\) odd numbers is given by:
\[ \text{Sum of odd numbers} = n^2 = 1008^2 = 1016064 \]

4. **Calculating the Even Sum**:
The even numbers from 2 to 2016 can be expressed as:
\[ 2, 4, 6, \ldots, 2016 \]
This also forms an arithmetic series where:
- First term \(a = 2\)
- Last term \(l = 2016\)
- Common difference \(d = 2\)

The number of terms \(n\) is:
\[ n = \frac{l - a}{d} + 1 = \frac{2016 - 2}{2} + 1 = 1008 \]

The sum of the first \(n\) even numbers is given by:
\[ \text{Sum of even numbers} = n(n + 1) = 1008 \times 1009 = 1017072 \]

5. **Combining the Results**:
Now we can substitute back into our expression for \(S\):
\[ S = \text{Sum of odd numbers} - \text{Sum of even numbers} = 1016064 - 1017072 = -1008 \]

### Final Answer:
Thus, the sum \(S\) is:
\[ \boxed{-1008} \]