(i) Find the sum of first n natural numbers.
(ii) Find the sum of first 100 natural numbers.

Answer

### Step-by-Step Solution

#### (i) Finding the Sum of the First n Natural Numbers

1. **Identify the Sequence**: The first n natural numbers are:
\[ 1, 2, 3, \ldots, n \]

2. **Recognize the Type of Sequence**: This sequence is an arithmetic progression (AP) where:
- The first term \( A = 1 \)
- The common difference \( D = 1 \)

3. **Use the Formula for the Sum of an AP**: The formula for the sum of the first n terms of an arithmetic progression is given by:
\[ S_n = \frac{n}{2} \times (2A + (n - 1)D) \]

4. **Substitute the Values**:
- Here, \( A = 1 \), \( D = 1 \), and we are looking for \( S_n \):
\[ S_n = \frac{n}{2} \times (2 \times 1 + (n - 1) \times 1) \]

5. **Simplify the Expression**:
- Calculate \( 2 \times 1 = 2 \)
- Calculate \( (n - 1) \times 1 = n - 1 \)
- Therefore:
\[ S_n = \frac{n}{2} \times (2 + n - 1) = \frac{n}{2} \times (n + 1) \]

6. **Final Result**: The sum of the first n natural numbers is:
\[ S_n = \frac{n(n + 1)}{2} \]

#### (ii) Finding the Sum of the First 100 Natural Numbers

1. **Set n = 100**: We want to find \( S_{100} \):
\[ S_{100} = \frac{100(100 + 1)}{2} \]

2. **Calculate the Expression**:
- Calculate \( 100 + 1 = 101 \)
- Therefore:
\[ S_{100} = \frac{100 \times 101}{2} \]

3. **Perform the Multiplication**:
- Calculate \( 100 \times 101 = 10100 \)

4. **Divide by 2**:
\[ S_{100} = \frac{10100}{2} = 5050 \]

5. **Final Result**: The sum of the first 100 natural numbers is:
\[ S_{100} = 5050 \]

### Summary of Results
- (i) The sum of the first n natural numbers is \( \frac{n(n + 1)}{2} \).
- (ii) The sum of the first 100 natural numbers is \( 5050 \).