Suppose for each `n in N, 1^(4)+2^(4)+.........n^(4)=an^(5)+bn^(4)+cn^(3)+dn^(2)+en+f`, then the value of b is

Options

A. `(1)/(5)`
B. `(1)/(2)`
C. `(1)/(3)`
D. 1

Explanation

To find the value of \( b \) in the equation

\[ 1^4 + 2^4 + \ldots + n^4 = a n^5 + b n^4 + c n^3 + d n^2 + e n + f, \]

we start with the known formula for the sum of the fourth powers of the first \( n \) natural numbers:

\[ \sum_{k=1}^{n} k^4 = \frac{n(n+1)(2n+1)(3n^2 + 3n - 1)}{30}. \]

### Step 1: Expand the formula

We need to expand the right-hand side of the formula.

\[ \sum_{k=1}^{n} k^4 = \frac{n(n+1)(2n+1)(3n^2 + 3n - 1)}{30}. \]

### Step 2: Simplify the expression

First, we can simplify the expression by multiplying the terms:

1. Calculate \( n(n+1) = n^2 + n \).
2. Calculate \( 2n + 1 \).
3. Calculate \( 3n^2 + 3n - 1 \).

Now, we multiply these together:

\[ n(n+1)(2n+1)(3n^2 + 3n - 1). \]

### Step 3: Collect like terms

After expanding the product, we will collect like terms to identify the coefficients of \( n^5, n^4, n^3, n^2, n, \) and the constant term.

### Step 4: Identify the coefficient of \( n^4 \)

To find \( b \), we need to focus on the coefficient of \( n^4 \) in the expanded polynomial.

After performing the multiplication and simplification, we will find the coefficient of \( n^4 \) in the final expression.

### Step 5: Divide by 30

Finally, since the entire expression is divided by 30, we will take the coefficient of \( n^4 \) from our expanded polynomial and divide it by 30 to find \( b \).

### Final Calculation

After performing the calculations, we find that the coefficient of \( n^4 \) is 15. Dividing this by 30 gives:

\[ b = \frac{15}{30} = \frac{1}{2}. \]

Thus, the value of \( b \) is

\[ \boxed{\frac{1}{2}}. \]