Find the value of 2551 S where `S = sum_(k=1)^(50) k/(k^(4)+k^(2)+1)`

Answer

To find the value of \( 2551 S \) where \( S = \sum_{k=1}^{50} \frac{k}{k^4 + k^2 + 1} \), we can follow these steps:

### Step 1: Simplify the Denominator
The denominator can be rewritten as:
\[ k^4 + k^2 + 1 = (k^2 + 1)^2 - k^2 \]
This can be factored further using the difference of squares:
\[ = (k^2 + 1 - k)(k^2 + 1 + k) \]
Thus, we have:
\[ k^4 + k^2 + 1 = (k^2 - k + 1)(k^2 + k + 1) \]

### Step 2: Rewrite the Sum
Now we can rewrite \( S \):
\[ S = \sum_{k=1}^{50} \frac{k}{(k^2 - k + 1)(k^2 + k + 1)} \]

### Step 3: Partial Fraction Decomposition
We will use partial fractions to break this down:
\[ \frac{k}{(k^2 - k + 1)(k^2 + k + 1)} = \frac{A}{k^2 - k + 1} + \frac{B}{k^2 + k + 1} \]
Multiplying through by the denominator:
\[ k = A(k^2 + k + 1) + B(k^2 - k + 1) \]
Expanding and combining like terms gives:
\[ k = (A + B)k^2 + (A - B)k + (A + B) \]
Setting coefficients equal, we have:
1. \( A + B = 0 \)
2. \( A - B = 1 \)
3. \( A + B = 0 \)

From \( A + B = 0 \), we can express \( B = -A \). Substituting into the second equation:
\[ A - (-A) = 1 \implies 2A = 1 \implies A = \frac{1}{2}, B = -\frac{1}{2} \]

### Step 4: Substitute Back into the Sum
Now substituting back, we have:
\[ S = \sum_{k=1}^{50} \left( \frac{1/2}{k^2 - k + 1} - \frac{1/2}{k^2 + k + 1} \right) \]
This can be simplified to:
\[ S = \frac{1}{2} \sum_{k=1}^{50} \left( \frac{1}{k^2 - k + 1} - \frac{1}{k^2 + k + 1} \right) \]

### Step 5: Telescoping Series
Notice that this is a telescoping series:
\[ S = \frac{1}{2} \left( \frac{1}{1} - \frac{1}{3} + \frac{1}{3} - \frac{1}{7} + \ldots + \frac{1}{2500} - \frac{1}{2551} \right) \]
Most terms cancel out, and we are left with:
\[ S = \frac{1}{2} \left( 1 - \frac{1}{2551} \right) \]
Calculating this gives:
\[ S = \frac{1}{2} \cdot \frac{2550}{2551} = \frac{1275}{2551} \]

### Step 6: Calculate \( 2551 S \)
Now we can find \( 2551 S \):
\[ 2551 S = 2551 \cdot \frac{1275}{2551} = 1275 \]

### Final Answer
Thus, the value of \( 2551 S \) is:
\[ \boxed{1275} \]