Read the statements carefully and select the correct option.
Statement-I : The sum of first n terms of an A.P. whose first term is A, the second term is B and the last term is L , is equal to `(( B + L -2A ) ( A + L) )/( 2 ( B - A) )`
Statement-II : If the sum of p terms of an A.P. is equal to the sum of its q terms, then the sum of its `(p + q)` terms is `p + q`.

Options

A. Both Statement-I and Statement-II are false.
B. Both Statement-I and Statement-II are true.
C. Statement-I is true but Statement-II is false.
D. Statement-I is false but Statement-II is true.

Answer

To solve the problem, we need to evaluate the two statements provided and determine their validity.

### Step-by-Step Solution:

**Statement I:**
The sum of the first \( n \) terms of an A.P. whose first term is \( A \), the second term is \( B \), and the last term is \( L \) is given by:
\[ S_n = \frac{(B + L - 2A)(A + L)}{2(B - A)} \]

1. **Identify the first term and common difference:**
- First term \( a = A \)
- Second term \( b = B \)
- Common difference \( d = B - A \)

2. **Identify the last term:**
- The last term \( L \) can be expressed in terms of \( n \):
\[ L = A + (n-1)d = A + (n-1)(B - A) \]

3. **Express \( n \) in terms of \( A \), \( B \), and \( L \):**
- Rearranging gives:
\[ L - A = (n-1)(B - A) \implies n = \frac{(L - A)}{(B - A)} + 1 \]

4. **Sum of the first \( n \) terms formula:**
- The sum of the first \( n \) terms of an A.P. is given by:
\[ S_n = \frac{n}{2} \times \text{(first term + last term)} = \frac{n}{2} \times (A + L) \]

5. **Substituting \( n \):**
- Substitute \( n \) into the sum formula:
\[ S_n = \frac{(L - A + (B - A))}{2(B - A)} \times (A + L) \]

6. **Simplifying the expression:**
- After simplification, we find that the expression matches the one given in Statement I.

**Conclusion for Statement I:**
- Statement I is **True**.

---

**Statement II:**
If the sum of \( p \) terms of an A.P. is equal to the sum of its \( q \) terms, then the sum of its \( (p + q) \) terms is \( p + q \).

1. **Using the sum formula for \( p \) and \( q \):**
- The sum of the first \( p \) terms:
\[ S_p = \frac{p}{2}(2A + (p-1)d) \]
- The sum of the first \( q \) terms:
\[ S_q = \frac{q}{2}(2A + (q-1)d) \]

2. **Setting the sums equal:**
- According to the statement:
\[ S_p = S_q \implies \frac{p}{2}(2A + (p-1)d) = \frac{q}{2}(2A + (q-1)d) \]

3. **Solving the equation:**
- Simplifying gives a relationship between \( p \) and \( q \) that does not lead to \( S_{p+q} = p + q \).

4. **Calculating \( S_{p+q} \):**
- The sum of the first \( p + q \) terms:
\[ S_{p+q} = \frac{(p+q)}{2}(2A + (p + q - 1)d) \]
- This expression does not simplify to \( p + q \).

**Conclusion for Statement II:**
- Statement II is **False**.

---

### Final Conclusion:
- **Statement I is True.**
- **Statement II is False.**