If `f(n)=sum_(r=1)^(n) r^(4)`, then the value of `sum_(r=1)^(n) r(n-r)^(3)` is equal to

Options

A. `(1)/(4){n^(2)(n+1)^(3)-4f(n)}`
B. `(1)/(4){n^(3)(n+1)^(2)-4f(n)}`
C. `(1)/(4){n^(2)(n+1)^(2)-4f(n)}`
D. none of these

Explanation

To solve the problem, we need to find the value of the summation \( \sum_{r=1}^{n} r(n-r)^3 \) given that \( f(n) = \sum_{r=1}^{n} r^4 \).

### Step-by-Step Solution:

1. **Rewrite the Expression**:
We start with the expression:
\[ S = \sum_{r=1}^{n} r(n-r)^3 \]

2. **Use the Property of Summation**:
We can use the property of summation that allows us to replace \( r \) with \( n - r \). This gives:
\[ S = \sum_{r=1}^{n} (n - r) r^3 \]

3. **Combine the Two Expressions**:
Now we can combine the two expressions:
\[ S = \sum_{r=1}^{n} r(n-r)^3 + \sum_{r=1}^{n} (n-r)r^3 \]
This can be simplified to:
\[ S = \sum_{r=1}^{n} \left[ r(n-r)^3 + (n-r)r^3 \right] \]

4. **Factor Out Common Terms**:
Notice that both terms can be factored:
\[ S = \sum_{r=1}^{n} \left[ r(n-r)^3 + n \cdot r^3 - r^4 \right] \]

5. **Simplify the Expression**:
We can rewrite the expression as:
\[ S = n \sum_{r=1}^{n} r^3 - \sum_{r=1}^{n} r^4 \]

6. **Use Known Formulas**:
We know the formulas for the summations:
\[ \sum_{r=1}^{n} r^3 = \left( \frac{n(n+1)}{2} \right)^2 \]
and \( f(n) = \sum_{r=1}^{n} r^4 \) is given.

7. **Substitute the Known Values**:
Substitute the known values into the expression:
\[ S = n \left( \frac{n(n+1)}{2} \right)^2 - f(n) \]

8. **Final Simplification**:
Now we can express \( S \) in a simplified form:
\[ S = \frac{n^2(n+1)^2}{4} - f(n) \]

9. **Final Result**:
Therefore, the value of \( \sum_{r=1}^{n} r(n-r)^3 \) is:
\[ S = \frac{n^2(n+1)^2}{4} - f(n) \]