Let's break down the problem and solve it step by step.

Problem:
If the sum of the first n even natural numbers is equal to k times the sum of the first n odd natural numbers, then what is the value of k?

Step 1: Understanding the problem
We need to find the value of k, which represents the ratio between the sum of the first n even natural numbers and the sum of the first n odd natural numbers.

Step 2: Expressing the sums mathematically
To find the sum of the first n even natural numbers, we use the formula:
Sum of even numbers = 2 + 4 + 6 + ... + 2n

Similarly, the sum of the first n odd natural numbers can be expressed as:
Sum of odd numbers = 1 + 3 + 5 + ... + (2n - 1)

Step 3: Calculating the sums
Using the formula for the sum of an arithmetic series, we can calculate the sum of the first n even numbers as:
Sum of even numbers = n * (first even number + last even number) / 2
= n * (2 + 2n) / 2
= n * (n + 1)

Similarly, the sum of the first n odd numbers can be calculated as:
Sum of odd numbers = n * (first odd number + last odd number) / 2
= n * (1 + 2n - 1) / 2
= n^2

Step 4: Equating the sums
According to the problem statement, the sum of the first n even numbers is equal to k times the sum of the first n odd numbers. Mathematically, this can be represented as:
n * (n + 1) = k * n^2

Simplifying the equation, we get:
n + 1 = k * n

Step 5: Finding the value of k
To find the value of k, we need to isolate it in the equation. Dividing both sides by n, we get:
1 + (1/n) = k

So, the value of k is 1 + (1/n), which corresponds to option (1).

Explanation for not choosing option (2)
Option (2) states that k = (n - 1)/n. However, if we substitute this value of k in the equation, we get:
n + 1 = (n - 1)/n * n
n + 1 = n - 1

This equation is not true for any value of n. Hence, option (2) is incorrect.

Therefore, the correct value of k is 1/n.

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