Let `a_1, a_2, a_3, ....a_n,.........`be in A.P. If `a_3 + a_7 + a_11 + a_15 = 72,` then the sum of itsfirst 17 terms is equal to :

Options

A. 153
B. 306
C. 612
D. 204

Explanation

To solve the problem step by step, we will follow the same logic as presented in the video transcript.

### Step-by-Step Solution:

1. **Understanding the A.P. Terms**:
In an arithmetic progression (A.P.), the nth term can be expressed as:
\[ a_n = a + (n - 1)d \]
where \( a \) is the first term and \( d \) is the common difference.

2. **Finding Specific Terms**:
We need to find the values of \( a_3, a_7, a_{11}, \) and \( a_{15} \):
- \( a_3 = a + (3 - 1)d = a + 2d \)
- \( a_7 = a + (7 - 1)d = a + 6d \)
- \( a_{11} = a + (11 - 1)d = a + 10d \)
- \( a_{15} = a + (15 - 1)d = a + 14d \)

3. **Setting Up the Equation**:
According to the problem, we have:
\[ a_3 + a_7 + a_{11} + a_{15} = 72 \]
Substituting the values we found:
\[ (a + 2d) + (a + 6d) + (a + 10d) + (a + 14d) = 72 \]
Simplifying this gives:
\[ 4a + (2d + 6d + 10d + 14d) = 72 \]
\[ 4a + 32d = 72 \]

4. **Solving for \( a + 8d \)**:
We can simplify the equation:
\[ 4a + 32d = 72 \]
Dividing the entire equation by 4:
\[ a + 8d = 18 \]

5. **Finding the Sum of the First 17 Terms**:
The formula for the sum of the first \( n \) terms of an A.P. is given by:
\[ S_n = \frac{n}{2} \times (2a + (n - 1)d) \]
For the first 17 terms:
\[ S_{17} = \frac{17}{2} \times (2a + 16d) \]
We can factor out 2 from \( 2a + 16d \):
\[ S_{17} = \frac{17}{2} \times 2(a + 8d) \]
Simplifying this gives:
\[ S_{17} = 17(a + 8d) \]
Since we found \( a + 8d = 18 \):
\[ S_{17} = 17 \times 18 = 306 \]

### Final Answer:
The sum of the first 17 terms of the A.P. is \( \boxed{306} \).