If x, x - 2 and 3x are in AP, then find the value of x

Answer

To solve the problem where \( x, x - 2, \) and \( 3x \) are in Arithmetic Progression (AP), we can follow these steps:

### Step 1: Understand the condition for AP
If three terms \( A, B, C \) are in AP, then the relation is given by:
\[ 2B = A + C \]
In our case, \( A = x \), \( B = x - 2 \), and \( C = 3x \).

### Step 2: Set up the equation
Using the AP condition, we can write:
\[ 2(x - 2) = x + 3x \]

### Step 3: Simplify the equation
Now simplify both sides:
\[ 2(x - 2) = 4x \]
Expanding the left side:
\[ 2x - 4 = 4x \]

### Step 4: Rearrange the equation
Now, we will rearrange the equation to isolate \( x \):
\[ 2x - 4 = 4x \]
Subtract \( 2x \) from both sides:
\[ -4 = 4x - 2x \]
This simplifies to:
\[ -4 = 2x \]

### Step 5: Solve for \( x \)
Now, divide both sides by 2:
\[ x = \frac{-4}{2} = -2 \]

### Final Answer
Thus, the value of \( x \) is:
\[ \boxed{-2} \]