Question

If \(-26\), x, 2 are in A.P., then the value of x is

• A. 14
• B. \(-13\)
• C. \(-12\)
• D. \(-14\)

Hint:

Whenever you need to find the middle term of three consecutive terms in an A.P., simply compute their average: \[ \text{Middle Term} = \frac{\text{First Term} + \text{Third Term}}{2} \] Here, \(\frac{-26 + 2}{2} = \frac{-24}{2} = -12\). This is highly efficient for exams!

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The question is from the topic of Arithmetic Progressions (A.P.).
We are given three terms, \(-26\), \(x\), and \(2\), which form a consecutive sequence in an Arithmetic Progression.
We need to calculate the value of the unknown middle term \(x\).

Step 2: Key Formula or Approach:
For any three numbers \(a\), \(b\), and \(c\) to be in an Arithmetic Progression, the difference between consecutive terms must be constant.
Thus: \[ b - a = c - b \] Rearranging this relationship, we find that the middle term \(b\) is the arithmetic mean of the first and third terms: \[ 2b = a + c \implies b = \frac{a + c}{2} \]

Step 3: Detailed Explanation:
1. Let the three given consecutive terms of the A.P. be: \[ a = -26 \] \[ b = x \] \[ c = 2 \] 2. Since these terms are in A.P., the common difference \(d\) must be equal: \[ x - (-26) = 2 - x \] \[ x + 26 = 2 - x \] 3. Bring all terms containing \(x\) to one side and constants to the other: \[ x + x = 2 - 26 \] \[ 2x = -24 \] 4. Solve for \(x\) by dividing by 2: \[ x = \frac{-24}{2} \] \[ x = -12 \]

Step 4: Final Answer:
The value of \(x\) is found to be \(-12\).
Therefore, the correct option is (C).