Statement 1 The difference between the sum of the first 100 even natural numbers and the sum of the first 100 odd natural numbers is 100.
Statement 2 The difference between the sum opf the first n even natural numbers and sum of the first n odd natural numbers is n.

Options

A. Statement 1 is true, Statement 2 is true, Statement 2 is a correct explanation for Statement 1
B. Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation for Statement 1
C. Statement 1 is true, Statement 2 is false
D. Statement 1 is false, Statement 2 is true

Explanation

To solve the problem step by step, we will analyze both statements and prove their validity.

### Step 1: Analyze Statement 2
Statement 2 claims that the difference between the sum of the first \( n \) even natural numbers and the sum of the first \( n \) odd natural numbers is \( n \).

#### Sum of the First \( n \) Even Natural Numbers
The first \( n \) even natural numbers are:
\[ 2, 4, 6, \ldots, 2n \]
The sum \( E \) of these numbers can be calculated using the formula:
\[ E = n \times \left(\frac{2 + 2n}{2}\right) = n \times (n + 1) = n(n + 1) \]

#### Sum of the First \( n \) Odd Natural Numbers
The first \( n \) odd natural numbers are:
\[ 1, 3, 5, \ldots, (2n - 1) \]
The sum \( O \) of these numbers can be calculated using the formula:
\[ O = n^2 \]

#### Difference Between the Sums
Now, we find the difference \( E - O \):
\[ E - O = n(n + 1) - n^2 = n \]
Thus, Statement 2 is true.

### Step 2: Analyze Statement 1
Statement 1 claims that the difference between the sum of the first 100 even natural numbers and the sum of the first 100 odd natural numbers is 100.

#### Calculate for \( n = 100 \)
Using the results from Statement 2, we can substitute \( n = 100 \):
\[ E = 100(100 + 1) = 100 \times 101 = 10100 \]
\[ O = 100^2 = 10000 \]
Now, we find the difference:
\[ E - O = 10100 - 10000 = 100 \]
Thus, Statement 1 is also true.

### Conclusion
Both statements are true, and Statement 2 provides a correct explanation for Statement 1.