Question

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

• A. 10^th
• B. 8^th
• C. 1^st
• D. None of these

Answer 1

The Correct Option is B

The correct option is (B): 8^th
Explanation:Let the arithmetic progression (AP) be represented as \( A_1, A_2, A_3, \ldots \) with the first term \( a \) and common difference \( d \). The \( n \)th term of an AP is given by \( A_n = a + (n-1)d \).
According to the problem, we need to equate the sums of specific terms:
\[A_6 + A_{15} = A_7 + A_{10} + A_{12}\]
This can be expressed as:
\[(a + 5d) + (a + 14d) = (a + 6d) + (a + 9d) + (a + 11d)\]
Simplifying both sides:
\[a + 19d = 3a + 26d\]
Rearranging the equation gives:
\[a + 19d - 3a - 26d = 0\]
This leads to:
\[-2a - 7d = 0 \quad \Rightarrow \quad 2a + 7d = 0\]
From this, we find:
\[a = -\frac{7}{2}d\]
Now, to determine the 8th term:
\[A_8 = a + 7d = -\frac{7}{2}d + 7d = \frac{7}{2}d\]
For \( A_8 \) to equal zero, \( d \) must be set to zero. Therefore, the 8th term \( A_8 \) is necessarily zero.
Thus, the answer is Option B: 8^th term.