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: Understand the property of AP.
In an A.P., the middle term of any three consecutive terms is the arithmetic mean of the other two.
Step 2: Apply the middle term property.
If \(-26\), \(x\), and \(2\) are in A.P., then \(x = \frac{-26 + 2}{2}\).
Step 3: Compute the mean.
\(x = \frac{-24}{2} = -12\).
Step 4: Verify using common difference.
\(d = x - (-26) = -12 + 26 = 14\) and \(d = 2 - x = 2 - (-12) = 14\). Both differences are equal, confirming an A.P.
Step 5: Check all options.
Option 3 gives \(x = -12\), which is correct.
Step 6: Select the correct option.
The value of \(x\) is \(-12\), option 3.
\[ \boxed{x = -12} \]