Question Stimulus :-
If 2x, x+8, 3x+1 are in A.P. Then the value of x will be -

Options

A. 3
B. 7
C. 5
D. `-2`

Answer

To find the value of \( x \) such that \( 2x, x + 8, 3x + 1 \) are in Arithmetic Progression (A.P.), we can follow these steps:

### Step 1: Understand the condition for A.P.
In an A.P., the difference between consecutive terms is constant. Therefore, we can set up the equation:
\[ a_2 - a_1 = a_3 - a_2 \]
where \( a_1 = 2x \), \( a_2 = x + 8 \), and \( a_3 = 3x + 1 \).

### Step 2: Write the equation based on the A.P. condition.
Substituting the values of \( a_1 \), \( a_2 \), and \( a_3 \) into the equation:
\[ (x + 8) - (2x) = (3x + 1) - (x + 8) \]

### Step 3: Simplify both sides of the equation.
On the left side:
\[ x + 8 - 2x = 8 - x \]
On the right side:
\[ 3x + 1 - x - 8 = 2x - 7 \]
Now we have:
\[ 8 - x = 2x - 7 \]

### Step 4: Solve for \( x \).
Now, rearranging the equation:
\[ 8 + 7 = 2x + x \]
This simplifies to:
\[ 15 = 3x \]
Dividing both sides by 3:
\[ x = 5 \]

### Conclusion:
The value of \( x \) is \( 5 \).

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