The sum of 8 terms of an A.P. is 64 and sum of 17 terms is 289. Find the sum of its 'n' terms.

Explanation

To solve the problem step by step, we will use the formula for the sum of the first n terms of an arithmetic progression (A.P.), which is given by:

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

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

### Step 1: Set up the equations based on the given information

We are given:
- The sum of the first 8 terms \( S_8 = 64 \)
- The sum of the first 17 terms \( S_{17} = 289 \)

Using the sum formula for the first 8 terms:

\[ S_8 = \frac{8}{2} \times (2A + (8-1)D) \]
\[ 64 = 4 \times (2A + 7D) \]
\[ 16 = 2A + 7D \] (Equation 1)

Now, using the sum formula for the first 17 terms:

\[ S_{17} = \frac{17}{2} \times (2A + (17-1)D) \]
\[ 289 = \frac{17}{2} \times (2A + 16D) \]
Multiplying both sides by 2 to eliminate the fraction:
\[ 578 = 17 \times (2A + 16D) \]
Dividing both sides by 17:
\[ 34 = 2A + 16D \] (Equation 2)

### Step 2: Solve the equations simultaneously

Now we have two equations:
1. \( 2A + 7D = 16 \) (Equation 1)
2. \( 2A + 16D = 34 \) (Equation 2)

We can subtract Equation 1 from Equation 2 to eliminate \( 2A \):

\[ (2A + 16D) - (2A + 7D) = 34 - 16 \]
\[ 9D = 18 \]
\[ D = 2 \]

### Step 3: Substitute \( D \) back to find \( A \)

Now that we have \( D = 2 \), we can substitute this value back into Equation 1 to find \( A \):

\[ 2A + 7(2) = 16 \]
\[ 2A + 14 = 16 \]
\[ 2A = 2 \]
\[ A = 1 \]

### Step 4: Find the sum of the first \( n \) terms

Now that we have both \( A \) and \( D \):
- \( A = 1 \)
- \( D = 2 \)

We can find the sum of the first \( n \) terms using the sum formula:

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

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

### Final Answer

The sum of the first \( n \) terms of the A.P. is:

\[ S_n = n^2 \]