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 find the sum of the first 17 terms of the arithmetic progression (A.P.), we start with the information given in the problem:

1. The sum of the first 8 terms (S8) is 64.
2. The sum of the first 15 terms (S15) is 225.

We will use the formula for the sum of the first n terms of an A.P., which is:

\[ S_n = \frac{n}{2} \times (2A + (n-1)D) \]

where:
- \( S_n \) is the sum of the first n terms,
- \( A \) is the first term,
- \( D \) is the common difference,
- \( n \) is the number of terms.

### Step 1: Set up the equations using the sum formula

For the first 8 terms:

\[ S_8 = \frac{8}{2} \times (2A + (8-1)D) = 64 \]

This simplifies to:

\[ 4 \times (2A + 7D) = 64 \]

Dividing both sides by 4:

\[ 2A + 7D = 16 \quad \text{(Equation 1)} \]

For the first 15 terms:

\[ S_{15} = \frac{15}{2} \times (2A + (15-1)D) = 225 \]

This simplifies to:

\[ \frac{15}{2} \times (2A + 14D) = 225 \]

Multiplying both sides by 2 to eliminate the fraction:

\[ 15 \times (2A + 14D) = 450 \]

Dividing both sides by 15:

\[ 2A + 14D = 30 \quad \text{(Equation 2)} \]

### Step 2: Solve the system of equations

Now we have two equations:

1. \( 2A + 7D = 16 \)
2. \( 2A + 14D = 30 \)

We can 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 3: Substitute D back to find A

Now, substitute \( D = 2 \) back 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 4: 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 further:

\[ S_{17} = \frac{17 \times 34}{2} = \frac{578}{2} = 289 \]

Thus, the sum of the first 17 terms is:

\[ \boxed{289} \]