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

Options

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

Answer

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 can follow these steps:

### Step 1: Identify the nth term
The nth term of the AP is given as:
\[ T_n = 3n + 7 \]

### Step 2: Find the first term (T_1) and the common difference (d)
To find the first term \( T_1 \):
\[ T_1 = 3(1) + 7 = 3 + 7 = 10 \]

To find the common difference \( d \), we can calculate \( T_2 - T_1 \):
\[ T_2 = 3(2) + 7 = 6 + 7 = 13 \]
\[ d = T_2 - T_1 = 13 - 10 = 3 \]

### Step 3: 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 arithmetic progression is:
\[ S_n = \frac{n}{2} \times (2a + (n-1)d) \]
where \( a \) is the first term, \( d \) is the common difference, and \( n \) is the number of terms.

### Step 4: Substitute the values for \( n = 50 \), \( a = 10 \), and \( d = 3 \)
Substituting the values into the formula:
\[ S_{50} = \frac{50}{2} \times (2 \times 10 + (50 - 1) \times 3) \]
\[ = 25 \times (20 + 49 \times 3) \]
\[ = 25 \times (20 + 147) \]
\[ = 25 \times 167 \]
\[ = 4175 \]

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