Understanding the problem:
The problem states that the sum of the first even natural numbers is equal to k times the sum of the first n odd natural numbers. We need to find the value of k.

Step-by-step solution:
To solve this problem, let's break it down into smaller steps:

Step 1:
Let's write down the sum of the first even natural numbers. The sum of the first n even natural numbers can be calculated using the formula: n(n+1). For example, the sum of the first 4 even natural numbers would be 4(4+1) = 20.

Step 2:
Next, let's write down the sum of the first n odd natural numbers. The sum of the first n odd natural numbers can be calculated using the formula: n^2. For example, the sum of the first 4 odd natural numbers would be 4^2 = 16.

Step 3:
According to the problem statement, the sum of the first even natural numbers is equal to k times the sum of the first n odd natural numbers. So, we can set up the equation:

n(n+1) = k * n^2

Step 4:
Let's simplify the equation:

n^2 + n = k * n^2

n^2 - k * n^2 + n = 0

Factoring out n from the equation:

n(n - k * n + 1) = 0

This equation will be true if either n = 0 or (n - k * n + 1) = 0.

Step 5:
If n = 0, the equation is satisfied. However, since n represents the number of terms, it cannot be zero.

So, let's solve the equation (n - k * n + 1) = 0.

n - k * n + 1 = 0

(1 - k) * n + 1 = 0

(1 - k) * n = -1

n = -1 / (1 - k)

Step 6:
Since n represents the number of terms, it must be a positive integer. Therefore, the equation n = -1 / (1 - k) cannot have a positive integer solution.

Step 7:
Hence, the value of k cannot be determined.