`underset(r=1)overset(n)sum ( r )/(r^(4)+r^(2)+1)` is equal to

Options

A. `(n^(2)+n)/(2(n^(2)+n+1))`
B. `(n^(2)+2n)/(2(n^(2)+n+1))`
C. `(2n^(2)+n)/(2(n^(2)+n+1))`
D. `(n^(2)+n)/((n^(2)+n+1))`

Answer

To solve the problem, we need to evaluate the summation:

\[ \sum_{r=1}^{n} \frac{r}{r^4 + r^2 + 1} \]

### Step 1: Simplifying the Denominator
First, we can rewrite the denominator \( r^4 + r^2 + 1 \). We can add and subtract \( r^2 \) to facilitate factoring:

\[ r^4 + r^2 + 1 = r^4 + 2r^2 + 1 - r^2 = (r^2 + 1)^2 - r^2 \]

This can be recognized as a difference of squares:

\[ (r^2 + 1)^2 - r^2 = (r^2 + 1 - r)(r^2 + 1 + r) \]

Thus, we have:

\[ r^4 + r^2 + 1 = (r^2 - r + 1)(r^2 + r + 1) \]

### Step 2: Rewrite the Summation
Now we can rewrite the summation:

\[ \sum_{r=1}^{n} \frac{r}{(r^2 - r + 1)(r^2 + r + 1)} \]

### Step 3: Partial Fraction Decomposition
Next, we can use partial fraction decomposition to express the fraction:

\[ \frac{r}{(r^2 - r + 1)(r^2 + r + 1)} = \frac{A}{r^2 - r + 1} + \frac{B}{r^2 + r + 1} \]

Multiplying through by the denominator gives:

\[ r = A(r^2 + r + 1) + B(r^2 - r + 1) \]

### Step 4: Finding Coefficients
Expanding the right-hand side:

\[ r = (A + B)r^2 + (A - B)r + (A + B) \]

Setting up equations by matching coefficients:

1. \( A + B = 0 \) (coefficient of \( r^2 \))
2. \( A - B = 1 \) (coefficient of \( r \))
3. \( A + B = 0 \) (constant term)

From the first equation, \( B = -A \). Substituting into the second equation:

\[ A - (-A) = 1 \implies 2A = 1 \implies A = \frac{1}{2} \]

Thus, \( B = -\frac{1}{2} \).

### Step 5: Rewrite the Summation
Now we can rewrite the summation using the coefficients found:

\[ \sum_{r=1}^{n} \left( \frac{1/2}{r^2 - r + 1} - \frac{1/2}{r^2 + r + 1} \right) \]

This simplifies to:

\[ \frac{1}{2} \left( \sum_{r=1}^{n} \frac{1}{r^2 - r + 1} - \sum_{r=1}^{n} \frac{1}{r^2 + r + 1} \right) \]

### Step 6: Evaluating the Summation
Now we can evaluate each of these sums. The sums will telescope when we compute them, leading to:

\[ \frac{1}{2} \left( \frac{1}{1} - \frac{1}{n^2 + n + 1} \right) \]

### Final Answer
Thus, the final result is:

\[ \frac{n(n+1)}{2(n^2 + n + 1)} \]