The sum of the 6th and 15th elements of an arithmetic progression is equal to the sum of 7th 10th and 12th elements of the same progression. Which elements of the series should necessarily be equal to zero

Options

A. 10th
B. 8th
C. 1st
D. 9th

Answer

To solve the problem, we need to find out which element of the arithmetic progression (AP) must necessarily be equal to zero given the condition about the sums of certain terms.

### Step-by-Step Solution:

1. **Define the Terms of the AP:**
Let the first term of the arithmetic progression be \( a \) and the common difference be \( d \). The \( n \)-th term of an AP can be expressed as:
\[ T_n = a + (n - 1)d \]

2. **Identify the Terms in the Problem:**
We need to find the 6th, 7th, 10th, 12th, and 15th terms:
- 6th term:
\[ T_6 = a + 5d \]
- 15th term:
\[ T_{15} = a + 14d \]
- 7th term:
\[ T_7 = a + 6d \]
- 10th term:
\[ T_{10} = a + 9d \]
- 12th term:
\[ T_{12} = a + 11d \]

3. **Set Up the Equation:**
According to the problem, the sum of the 6th and 15th terms is equal to the sum of the 7th, 10th, and 12th terms:
\[ T_6 + T_{15} = T_7 + T_{10} + T_{12} \]
Substituting the terms:
\[ (a + 5d) + (a + 14d) = (a + 6d) + (a + 9d) + (a + 11d) \]

4. **Simplify the Equation:**
Combine like terms:
\[ 2a + 19d = 3a + 26d \]

5. **Rearranging the Equation:**
Move all terms involving \( a \) to one side and terms involving \( d \) to the other side:
\[ 2a - 3a = 26d - 19d \]
This simplifies to:
\[ -a = 7d \]
or
\[ a + 7d = 0 \]

6. **Identify the Zero Term:**
We can express \( a \) in terms of \( d \):
\[ a = -7d \]
Now, we need to find which term is zero. The \( n \)-th term can be expressed as:
\[ T_n = a + (n - 1)d \]
Substituting \( a = -7d \):
\[ T_n = -7d + (n - 1)d = (n - 8)d \]
For \( T_n \) to be zero:
\[ (n - 8)d = 0 \]
This implies either \( d = 0 \) (which is not the case since it would make all terms equal) or \( n - 8 = 0 \), giving us \( n = 8 \).

### Conclusion:
The 8th term of the arithmetic progression must necessarily be equal to zero.