The sum of first 8 terms of an A.P. is 64 and that of first 15 terms is 225. Find the sum of its first 17 terms.

Answer

To solve the problem, we need to find the sum of the first 17 terms of an arithmetic progression (A.P.) given that the sum of the first 8 terms is 64 and the sum of the first 15 terms is 225.

### Step 1: Write the formula for the sum of n terms of an A.P.
The sum of the first n terms (S_n) of an A.P. can be calculated using the formula:
\[ S_n = \frac{n}{2} \times (2A + (n-1)d) \]
where \( A \) is the first term and \( d \) is the common difference.

### Step 2: Set up the equations based on the given information.
1. For the first 8 terms:
\[ S_8 = \frac{8}{2} \times (2A + (8-1)d) = 64 \]
Simplifying this gives:
\[ 4 \times (2A + 7d) = 64 \]
Dividing both sides by 4:
\[ 2A + 7d = 16 \quad \text{(Equation 1)} \]

2. For the first 15 terms:
\[ S_{15} = \frac{15}{2} \times (2A + (15-1)d) = 225 \]
Simplifying this gives:
\[ \frac{15}{2} \times (2A + 14d) = 225 \]
Multiplying both sides by 2:
\[ 15 \times (2A + 14d) = 450 \]
Dividing both sides by 15:
\[ 2A + 14d = 30 \quad \text{(Equation 2)} \]

### Step 3: Solve the system of equations.
Now we have the following two equations:
1. \( 2A + 7d = 16 \) (Equation 1)
2. \( 2A + 14d = 30 \) (Equation 2)

Subtract Equation 1 from Equation 2:
\[ (2A + 14d) - (2A + 7d) = 30 - 16 \]
This simplifies to:
\[ 7d = 14 \]
Dividing both sides by 7:
\[ d = 2 \]

### Step 4: Substitute the value of d back into one of the equations.
Substituting \( d = 2 \) into Equation 1:
\[ 2A + 7(2) = 16 \]
This simplifies to:
\[ 2A + 14 = 16 \]
Subtracting 14 from both sides:
\[ 2A = 2 \]
Dividing both sides by 2:
\[ A = 1 \]

### Step 5: Find the sum of the first 17 terms.
Now that we have \( A = 1 \) and \( d = 2 \), we can find the sum of the first 17 terms:
\[ S_{17} = \frac{17}{2} \times (2A + (17-1)d) \]
Substituting the values of \( A \) and \( d \):
\[ S_{17} = \frac{17}{2} \times (2(1) + 16(2)) \]
This simplifies to:
\[ S_{17} = \frac{17}{2} \times (2 + 32) = \frac{17}{2} \times 34 \]
Calculating this gives:
\[ S_{17} = \frac{17 \times 34}{2} = \frac{578}{2} = 289 \]

### Final Answer:
The sum of the first 17 terms of the A.P. is **289**.