If 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

Answer

To find the sum of the first 105 terms of an arithmetic progression (AP) where the nth term is given by the formula \( a_n = \frac{3+n}{4} \), we can follow these steps:

### Step 1: Identify the first term
To find the first term \( a_1 \), we substitute \( n = 1 \) into the formula for the nth term:
\[ a_1 = \frac{3 + 1}{4} = \frac{4}{4} = 1 \]

### Step 2: Identify the last term
Next, we need to find the 105th term \( a_{105} \) by substituting \( n = 105 \) into the formula:
\[ a_{105} = \frac{3 + 105}{4} = \frac{108}{4} = 27 \]

### Step 3: Use the formula for the sum of the first n terms
The formula for the sum of the first \( n \) terms of an AP is given by:
\[ S_n = \frac{n}{2} \times (a_1 + a_n) \]
In our case, \( n = 105 \), \( a_1 = 1 \), and \( a_{105} = 27 \).

### Step 4: Substitute the values into the formula
Now we can substitute the values into the sum formula:
\[ S_{105} = \frac{105}{2} \times (1 + 27) = \frac{105}{2} \times 28 \]

### Step 5: Calculate the sum
Calculating further:
\[ S_{105} = \frac{105 \times 28}{2} = \frac{2940}{2} = 1470 \]

### Final Answer
Thus, the sum of the first 105 terms is:
\[ \boxed{1470} \]