The nth term of an AP. Is `(3+n)/(4)`, then the sum of first 105 terms is

Options

A. 270
B. 735
C. 1409
D. 1470

Explanation

To solve the problem, we need to find the sum of the first 105 terms of the arithmetic progression (AP) given that the nth term of the AP is \( a_n = \frac{3+n}{4} \).

### Step-by-Step Solution:

1. **Identify the nth term**:
The nth term of the AP is given as:
\[ a_n = \frac{3+n}{4} \]

2. **Find the first term (a)**:
To find the first term \( a_1 \), we substitute \( n = 1 \) into the nth term formula:
\[ a_1 = \frac{3+1}{4} = \frac{4}{4} = 1 \]

3. **Find the last term (l)**:
To find the last term \( a_{105} \), we substitute \( n = 105 \) into the nth term formula:
\[ a_{105} = \frac{3+105}{4} = \frac{108}{4} = 27 \]

4. **Use the sum formula for an AP**:
The formula for the sum of the first \( n \) terms of an AP is:
\[ S_n = \frac{n}{2} \times (a + l) \]
Here, \( n = 105 \), \( a = 1 \), and \( l = 27 \).

5. **Substitute the values into the formula**:
\[ S_{105} = \frac{105}{2} \times (1 + 27) = \frac{105}{2} \times 28 \]

6. **Calculate the sum**:
\[ S_{105} = \frac{105 \times 28}{2} = \frac{2940}{2} = 1470 \]

Thus, the sum of the first 105 terms of the AP is \( \boxed{1470} \).