If the nth term of an arithmetic progression is `3n+7,` then what is the sum of its first 50 terms ?

Options

A. 3925
B. 4100
C. 4175
D. 8200

Explanation

To find the sum of the first 50 terms of the arithmetic progression (AP) where the nth term is given by \( T_n = 3n + 7 \), we will follow these steps:

### Step 1: Identify the first term and the common difference
The nth term of the AP is given by:
\[ T_n = 3n + 7 \]
To find the first term \( T_1 \):
\[ T_1 = 3(1) + 7 = 3 + 7 = 10 \]
To find the second term \( T_2 \):
\[ T_2 = 3(2) + 7 = 6 + 7 = 13 \]
The common difference \( d \) can be calculated as:
\[ d = T_2 - T_1 = 13 - 10 = 3 \]

### Step 2: Use the formula for the sum of the first n terms of an AP
The formula for the sum of the first \( n \) terms \( S_n \) of an AP is given by:
\[ S_n = \frac{n}{2} \times (2a + (n - 1)d) \]
where \( a \) is the first term and \( d \) is the common difference.

### Step 3: Substitute the values into the formula
Here, \( n = 50 \), \( a = 10 \), and \( d = 3 \). Plugging these values into the formula:
\[ S_{50} = \frac{50}{2} \times (2 \times 10 + (50 - 1) \times 3) \]
Calculating further:
\[ S_{50} = 25 \times (20 + 49 \times 3) \]
Calculating \( 49 \times 3 \):
\[ 49 \times 3 = 147 \]
So,
\[ S_{50} = 25 \times (20 + 147) = 25 \times 167 \]

### Step 4: Calculate the final sum
Now, calculate \( 25 \times 167 \):
\[ 25 \times 167 = 4175 \]

### Conclusion
The sum of the first 50 terms of the arithmetic progression is:
\[ \boxed{4175} \]