In any A.P. if sum of first six terms is 5 times the sum of next six terms then which term is zero?

Options

A. 10 th
B. 11 th
C. 12 th
D. 13 th

Explanation

To solve the problem step by step, we will use the properties of an arithmetic progression (A.P.) and the formulas for the sum of terms in an A.P.

### Step 1: Understand the problem
We are given that the sum of the first six terms of an A.P. is five times the sum of the next six terms. We need to find which term in the A.P. is zero.

### Step 2: Define the terms of the A.P.
Let:
- \( A \) = first term of the A.P.
- \( D \) = common difference of the A.P.

### Step 3: Write the formula for the sum of the first \( n \) terms of an A.P.
The sum \( S_n \) of the first \( n \) terms of an A.P. is given by the formula:
\[ S_n = \frac{n}{2} \left( 2A + (n - 1)D \right) \]

### Step 4: Calculate the sum of the first six terms
Using the formula for \( n = 6 \):
\[ S_6 = \frac{6}{2} \left( 2A + (6 - 1)D \right) = 3(2A + 5D) = 6A + 15D \]

### Step 5: Calculate the sum of the next six terms
The next six terms are the 7th to the 12th terms. The sum of the first 12 terms \( S_{12} \) can be calculated as:
\[ S_{12} = \frac{12}{2} \left( 2A + (12 - 1)D \right) = 6(2A + 11D) = 12A + 66D \]
Thus, the sum of the next six terms \( S_{7 \text{ to } 12} \) is:
\[ S_{7 \text{ to } 12} = S_{12} - S_6 = (12A + 66D) - (6A + 15D) = 6A + 51D \]

### Step 6: Set up the equation based on the problem statement
According to the problem, we have:
\[ S_6 = 5 \times S_{7 \text{ to } 12} \]
Substituting the sums we calculated:
\[ 6A + 15D = 5(6A + 51D) \]

### Step 7: Expand and simplify the equation
Expanding the right side:
\[ 6A + 15D = 30A + 255D \]
Now, rearranging gives:
\[ 6A - 30A + 15D - 255D = 0 \]
This simplifies to:
\[ -24A - 240D = 0 \]
Dividing through by -24:
\[ A + 10D = 0 \]

### Step 8: Find the term that is zero
The \( n \)-th term of an A.P. is given by:
\[ T_n = A + (n - 1)D \]
To find when this term is zero, we set:
\[ A + (n - 1)D = 0 \]
Substituting \( A = -10D \):
\[ -10D + (n - 1)D = 0 \]
This simplifies to:
\[ (n - 11)D = 0 \]
Thus, \( n - 11 = 0 \) or \( D = 0 \). Since \( D \) cannot be zero in a non-degenerate A.P., we have:
\[ n = 11 \]
Therefore, the 11th term is zero.

### Final Answer
The term that is zero is the **11th term**.

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