In an A.P. five times the fifth term is equal tyo eight times thte eight term. Then the sum of the first twenty five terms is equal to :

Options

A. `25`
B. `25/2`
C. `-25`
D. `0`

Answer

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

### Step 1: Understand the problem
We are given that five times the fifth term of an arithmetic progression (A.P.) is equal to eight times the eighth term. We need to find the sum of the first 25 terms of this A.P.

### Step 2: Write the formula for the nth term
The nth term of an A.P. is given by the formula:
\[ A_n = A + (n-1)D \]
where \( A \) is the first term and \( D \) is the common difference.

### Step 3: Express the 5th and 8th terms
Using the formula:
- The 5th term \( A_5 \) is:
\[ A_5 = A + (5-1)D = A + 4D \]
- The 8th term \( A_8 \) is:
\[ A_8 = A + (8-1)D = A + 7D \]

### Step 4: Set up the equation based on the given condition
According to the problem:
\[ 5 \times A_5 = 8 \times A_8 \]
Substituting the expressions for \( A_5 \) and \( A_8 \):
\[ 5(A + 4D) = 8(A + 7D) \]

### Step 5: Expand and simplify the equation
Expanding both sides:
\[ 5A + 20D = 8A + 56D \]

Now, rearranging the equation to isolate terms:
\[ 5A - 8A = 56D - 20D \]
\[ -3A = 36D \]

### Step 6: Solve for A in terms of D
Dividing both sides by -3:
\[ A = -12D \]

### Step 7: Find the sum of the first 25 terms
The formula for the sum of the first \( n \) terms of an A.P. is:
\[ S_n = \frac{n}{2} \times (2A + (n-1)D) \]
For \( n = 25 \):
\[ S_{25} = \frac{25}{2} \times (2A + (25-1)D) \]
Substituting \( A = -12D \):
\[ S_{25} = \frac{25}{2} \times (2(-12D) + 24D) \]
\[ S_{25} = \frac{25}{2} \times (-24D + 24D) \]
\[ S_{25} = \frac{25}{2} \times 0 \]
\[ S_{25} = 0 \]

### Final Answer
The sum of the first 25 terms is:
\[ S_{25} = 0 \]