If `8^(th)` of an A.P. is 15 , then the sum of first
15 terms is

Options

A. `180`
B. 210
C. 225
D. 240

Answer

To solve the problem step by step, we will use the formulas related to arithmetic progressions (A.P.).

### Step 1: Understand the given information
We are given that the 8th term of an A.P. is 15. The formula for the nth term of an A.P. is given by:
\[ a_n = a + (n-1)d \]
where \( a \) is the first term, \( d \) is the common difference, and \( n \) is the term number.

### Step 2: Write the equation for the 8th term
For the 8th term, we have:
\[ a_8 = a + (8-1)d = a + 7d \]
According to the problem, \( a_8 = 15 \). Therefore, we can write:
\[ a + 7d = 15 \quad \text{(1)} \]

### 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 A.P. is given by:
\[ S_n = \frac{n}{2} \times (2a + (n-1)d) \]
We need to find the sum of the first 15 terms, so we set \( n = 15 \):
\[ S_{15} = \frac{15}{2} \times (2a + (15-1)d) = \frac{15}{2} \times (2a + 14d) \]

### Step 4: Simplify the sum formula
We can factor out 2 from the expression:
\[ S_{15} = \frac{15}{2} \times (2a + 14d) = \frac{15}{2} \times 2 \left( a + 7d \right) = 15 \times (a + 7d) \]

### Step 5: Substitute the value from equation (1)
From equation (1), we know that \( a + 7d = 15 \). Therefore, we can substitute this into our sum formula:
\[ S_{15} = 15 \times (a + 7d) = 15 \times 15 = 225 \]

### Conclusion
Thus, the sum of the first 15 terms of the A.P. is:
\[ \boxed{225} \]