Let `f(n)=Sigma_(r=1)^(10n)(6+rd) and g(n)=Sigma_(r=1)^(n)(6+rd)`, where `n in N, d ne 0.` If `(f(n))/(g(n))` is independent of n, then d is equal to

Options

A. 12
B. `-6`
C. 6
D. `-12`

Answer

To solve the problem, we need to find the value of \( d \) such that the ratio \( \frac{f(n)}{g(n)} \) is independent of \( n \). Let's break down the problem step by step.

### Step 1: Define \( f(n) \) and \( g(n) \)

We have:
\[ f(n) = \sum_{r=1}^{10n} (6 + rd) \]
\[ g(n) = \sum_{r=1}^{n} (6 + rd) \]

### Step 2: Simplify \( f(n) \)

To simplify \( f(n) \):
\[ f(n) = \sum_{r=1}^{10n} 6 + \sum_{r=1}^{10n} rd \]
The first term can be simplified as:
\[ \sum_{r=1}^{10n} 6 = 6 \cdot 10n = 60n \]
The second term can be simplified using the formula for the sum of the first \( m \) natural numbers:
\[ \sum_{r=1}^{10n} r = \frac{10n(10n + 1)}{2} \]
Thus,
\[ \sum_{r=1}^{10n} rd = d \cdot \frac{10n(10n + 1)}{2} \]
Combining these results, we have:
\[ f(n) = 60n + d \cdot \frac{10n(10n + 1)}{2} \]

### Step 3: Simplify \( g(n) \)

Now, simplify \( g(n) \):
\[ g(n) = \sum_{r=1}^{n} 6 + \sum_{r=1}^{n} rd \]
The first term simplifies to:
\[ \sum_{r=1}^{n} 6 = 6n \]
The second term simplifies as:
\[ \sum_{r=1}^{n} r = \frac{n(n + 1)}{2} \]
Thus,
\[ \sum_{r=1}^{n} rd = d \cdot \frac{n(n + 1)}{2} \]
Combining these results, we have:
\[ g(n) = 6n + d \cdot \frac{n(n + 1)}{2} \]

### Step 4: Form the Ratio \( \frac{f(n)}{g(n)} \)

Now, we can form the ratio:
\[ \frac{f(n)}{g(n)} = \frac{60n + d \cdot \frac{10n(10n + 1)}{2}}{6n + d \cdot \frac{n(n + 1)}{2}} \]

### Step 5: Analyze Independence of \( n \)

For \( \frac{f(n)}{g(n)} \) to be independent of \( n \), the degree of \( n \) in the numerator must equal the degree of \( n \) in the denominator.

- The numerator simplifies to:
\[ 60n + d \cdot 50n^2 + \frac{d \cdot 10n}{2} \]
- The denominator simplifies to:
\[ 6n + \frac{d}{2}n^2 + \frac{d}{2}n \]

### Step 6: Set the Coefficients

To ensure independence from \( n \), we equate the coefficients of \( n^2 \) in both the numerator and denominator:
\[ d \cdot 50 = \frac{d}{2} \]

### Step 7: Solve for \( d \)

Solving the equation:
\[ 50d = \frac{d}{2} \]
Multiplying both sides by 2 gives:
\[ 100d = d \]
Rearranging gives:
\[ 99d = 0 \]
Since \( d \neq 0 \), we can conclude:
\[ d = -12 \]

### Final Answer

Thus, the value of \( d \) is:
\[ \boxed{-12} \]