If `S=sum_(r=1)^(80)(r)/((r^4+r^2+1))` , then the value of `(6481s)/(1000)` is

Answer

To solve the problem, we need to evaluate the sum

\[ S = \sum_{r=1}^{80} \frac{r}{r^4 + r^2 + 1} \]

and then find the value of \(\frac{6481S}{1000}\).

### Step 1: Simplifying the Denominator

The denominator can be rewritten:

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

This is a difference of squares, which can be factored as:

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

### Step 2: Rewrite the Sum

Now we can rewrite the sum \(S\):

\[ S = \sum_{r=1}^{80} \frac{r}{(r^2 - r + 1)(r^2 + r + 1)} \]

### Step 3: Partial Fraction Decomposition

We can use partial fractions to separate the terms:

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

Multiplying through by the denominator:

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

### Step 4: Solve for A and B

Expanding the right-hand side:

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

Setting coefficients equal gives us a system of equations:

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:

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

### Step 5: Rewrite S

Now we can rewrite \(S\):

\[ S = \sum_{r=1}^{80} \left( \frac{1/2}{r^2 - r + 1} - \frac{1/2}{r^2 + r + 1} \right) \]

This simplifies to:

\[ S = \frac{1}{2} \sum_{r=1}^{80} \left( \frac{1}{r^2 - r + 1} - \frac{1}{r^2 + r + 1} \right) \]

### Step 6: Telescoping Series

Notice that this is a telescoping series. Most terms will cancel out. Evaluating the first few and the last few terms will help us find the remaining terms.

### Step 7: Evaluate the Remaining Terms

The first term when \(r=1\) is:

\[ \frac{1}{1^2 - 1 + 1} = 1 \]

The last term when \(r=80\) is:

\[ \frac{1}{80^2 + 80 + 1} = \frac{1}{6481} \]

Thus, we can find \(S\):

\[ S = \frac{1}{2} \left( 1 - \frac{1}{6481} \right) = \frac{1}{2} \left( \frac{6480}{6481} \right) = \frac{6480}{12962} \]

### Step 8: Calculate \(\frac{6481S}{1000}\)

Now substituting \(S\) into \(\frac{6481S}{1000}\):

\[ \frac{6481 \cdot \frac{6480}{12962}}{1000} = \frac{6481 \cdot 6480}{12962000} \]

Calculating this gives:

\[ = \frac{6481 \cdot 6480}{12962000} = 3.24 \]

### Final Answer

Thus, the value of \(\frac{6481S}{1000}\) is:

\[ \boxed{3.24} \]