If the first term of an A.P. is 100 and sum of its first 6 terms is 5 times the sum of next 6 terms, then find the common difference of the A.P.

Explanation

To solve the problem step-by-step, we will follow the reasoning laid out in the video transcript.

### Step 1: Identify the given information
We know that:
- The first term \( A = 100 \)
- The sum of the first 6 terms is 5 times the sum of the next 6 terms.

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

### Step 3: Calculate \( S_6 \) and \( S_{12} \)
For the first 6 terms:
\[ S_6 = \frac{6}{2} \times (2 \times 100 + (6-1)D) = 3 \times (200 + 5D) = 600 + 15D \]

For the first 12 terms:
\[ S_{12} = \frac{12}{2} \times (2 \times 100 + (12-1)D) = 6 \times (200 + 11D) = 1200 + 66D \]

### Step 4: Set up the equation based on the problem statement
According to the problem, the sum of the first 6 terms is 5 times the sum of the next 6 terms:
\[ S_6 = 5 \times (S_{12} - S_6) \]
Substituting the expressions we found:
\[ 600 + 15D = 5 \times ((1200 + 66D) - (600 + 15D)) \]

### Step 5: Simplify the equation
First, simplify the right side:
\[ 600 + 15D = 5 \times (1200 + 66D - 600 - 15D) \]
\[ 600 + 15D = 5 \times (600 + 51D) \]
\[ 600 + 15D = 3000 + 255D \]

### Step 6: Rearrange the equation
Now, let's rearrange the equation:
\[ 600 + 15D - 255D = 3000 \]
\[ 600 - 240D = 3000 \]
\[ -240D = 3000 - 600 \]
\[ -240D = 2400 \]

### Step 7: Solve for \( D \)
Now, divide both sides by -240:
\[ D = \frac{2400}{-240} = -10 \]

### Conclusion
The common difference \( D \) of the A.P. is \( -10 \).

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