If `mgt1` and `ninN`, such that
`1^(m)+2^(m)+3^(m)+...+n^(m)gtn((n+1)/(k))^(m)` Then, k=

Options

A. 2
B. n
C. m
D. 1

Explanation

To solve the given inequality \( 1^m + 2^m + 3^m + \ldots + n^m > n \left( \frac{n+1}{k} \right)^m \), we will follow the steps outlined below.

### Step 1: Understanding the Arithmetic Mean Inequality
We know from the Arithmetic Mean - Geometric Mean (AM-GM) inequality that:
\[ \frac{1^m + 2^m + 3^m + \ldots + n^m}{n} \geq \left( \frac{1 + 2 + 3 + \ldots + n}{n} \right)^m \]
This means that the arithmetic mean of the \( m \)-th powers of the first \( n \) natural numbers is greater than or equal to the \( m \)-th power of the arithmetic mean of the first \( n \) natural numbers.

### Step 2: Calculate the Arithmetic Mean of the First \( n \) Natural Numbers
The sum of the first \( n \) natural numbers is given by:
\[ 1 + 2 + 3 + \ldots + n = \frac{n(n + 1)}{2} \]
Thus, the arithmetic mean is:
\[ \frac{1 + 2 + 3 + \ldots + n}{n} = \frac{n(n + 1)/2}{n} = \frac{n + 1}{2} \]

### Step 3: Substitute into the AM-GM Inequality
Substituting this back into our inequality gives:
\[ 1^m + 2^m + 3^m + \ldots + n^m > n \left( \frac{n + 1}{2} \right)^m \]

### Step 4: Rearranging the Inequality
We can rearrange our inequality to compare it with the original inequality:
\[ 1^m + 2^m + 3^m + \ldots + n^m > n \left( \frac{n + 1}{k} \right)^m \]
This implies:
\[ n \left( \frac{n + 1}{2} \right)^m > n \left( \frac{n + 1}{k} \right)^m \]

### Step 5: Canceling \( n \) (Assuming \( n > 0 \))
Since \( n \) is a natural number (and thus greater than 0), we can safely divide both sides by \( n \):
\[ \left( \frac{n + 1}{2} \right)^m > \left( \frac{n + 1}{k} \right)^m \]

### Step 6: Taking the \( m \)-th Root
Taking the \( m \)-th root of both sides (since \( m > 1 \)):
\[ \frac{n + 1}{2} > \frac{n + 1}{k} \]

### Step 7: Cross-Multiplying
Cross-multiplying gives:
\[ (n + 1)k > 2(n + 1) \]
Assuming \( n + 1 > 0 \), we can divide both sides by \( n + 1 \):
\[ k > 2 \]

### Step 8: Finding the Value of \( k \)
Since we need to find the value of \( k \) such that the original inequality holds, we can conclude that:
\[ k = 2 \]

Thus, the value of \( k \) is \( 2 \).

### Final Answer
\[ k = 2 \]