Assertion (A) : Sum of all 11 terms of an A.P whose middle most term is 30 is 330.
Reason (R ) : Sum of first n terms of an A.P is given by `S_(n)=(n)/(2)[a+l]`, `l` is the middle 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 problem, we will analyze the assertion and the reason step by step.

### Step 1: Understand the Assertion
The assertion states that the sum of all 11 terms of an A.P. (Arithmetic Progression) whose middlemost term is 30 is 330.

### Step 2: Identify the Middle Term
In an A.P. with 11 terms, the middle term is the 6th term. Therefore, if the middle term (T6) is given as 30, we can express this as:
\[ T_6 = a + 5d = 30 \]
where \( a \) is the first term and \( d \) is the common difference.

### Step 3: Calculate the Sum of the 11 Terms
The formula for the sum of the first \( n \) terms of an A.P. is:
\[ S_n = \frac{n}{2} \times (a + l) \]
where \( l \) is the last term. For 11 terms, we have:
\[ S_{11} = \frac{11}{2} \times (a + l) \]

### Step 4: Express the Last Term
The last term \( l \) can be expressed as:
\[ l = a + 10d \]
Thus, we can rewrite the sum as:
\[ S_{11} = \frac{11}{2} \times (a + (a + 10d)) \]
\[ S_{11} = \frac{11}{2} \times (2a + 10d) \]
\[ S_{11} = \frac{11}{2} \times 2(a + 5d) \]
\[ S_{11} = 11(a + 5d) \]

### Step 5: Substitute the Middle Term
Since we know from the assertion that \( a + 5d = 30 \), we can substitute this into the sum:
\[ S_{11} = 11 \times 30 = 330 \]

### Conclusion
The assertion is correct: the sum of all 11 terms of the A.P. is indeed 330.

### Step 6: Analyze the Reason
The reason states that the sum of the first \( n \) terms of an A.P. is given by \( S_n = \frac{n}{2} (a + l) \), but it incorrectly claims that \( l \) is the middle term. In fact, \( l \) is the last term of the A.P., not the middle term.

### Final Answer
- Assertion (A) is true.
- Reason (R) is false.