Assertion (A) : Sum of first hundred even natural numbers divisible by 5 is 500 .
Reason (R ) : Sum of first n terms of an A.P is given by `S_(N)=(n)/(2)[a+l],l` is last term

Options

A. Both A and R are true and R is the correct explanation for A.
B. Both A and R are true and R is not correct explanation for A.
C. A is true but R is false.
D. A is false but R is true.

Answer

To solve the question, we need to evaluate the assertion and the reason provided.

**Assertion (A)**: The sum of the first hundred even natural numbers divisible by 5 is 500.

**Reason (R)**: The sum of the first n terms of an A.P. is given by \( S_n = \frac{n}{2} (a + l) \), where \( l \) is the last term.

### Step-by-Step Solution:

1. **Identify the first hundred even natural numbers**:
The first hundred even natural numbers are:
2, 4, 6, 8, ..., 200.

2. **Identify the even natural numbers that are divisible by 5**:
The even natural numbers divisible by 5 can be expressed as:
10, 20, 30, ..., up to the 100th term.

3. **Determine the sequence**:
This forms an arithmetic progression (A.P.) where:
- The first term \( a = 10 \)
- The common difference \( d = 10 \)
- The number of terms \( n = 100 \)

4. **Find the last term (l)**:
The last term \( l \) of the first 100 terms can be calculated as:
\[ l = a + (n - 1) \cdot d = 10 + (100 - 1) \cdot 10 = 10 + 990 = 1000 \]

5. **Calculate the sum of the first 100 terms (S100)**:
Using the formula for the sum of an A.P.:
\[ S_n = \frac{n}{2} (a + l) \]
Plugging in the values:
\[ S_{100} = \frac{100}{2} (10 + 1000) = 50 \cdot 1010 = 50500 \]

6. **Conclusion about the assertion**:
The assertion states that the sum is 500, which is incorrect. The correct sum is 50500.

7. **Conclusion about the reason**:
The reason provided is the correct formula for the sum of the first n terms of an A.P., which is true.

### Final Evaluation:
- Assertion (A) is **false**.
- Reason (R) is **true**.