Question

The value of x for which \( 2x, (x + 10) \) and \( (3x + 2) \) are the three consecutive terms of an A.P. is :

• A. 6
• B. - 6
• C. 18
• D. - 18

Hint:

The middle term of three consecutive A.P. terms is the arithmetic mean of the first and third terms. Use \( \text{Middle} = \frac{\text{First} + \text{Third}}{2} \) for quick solving.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
In an Arithmetic Progression (A.P.), the difference between any two consecutive terms is constant.
Step 2: Key Formula or Approach:
If \( a, b, c \) are in A.P., then \( b - a = c - b \), or more simply: \( 2b = a + c \).
Step 3: Detailed Explanation:
Let \( a = 2x, b = x + 10, c = 3x + 2 \).
Using the property \( 2b = a + c \):
\[ 2(x + 10) = 2x + (3x + 2) \]
Expand the left side and combine terms on the right:
\[ 2x + 20 = 5x + 2 \]
Shift terms involving \( x \) to one side and constants to the other:
\[ 20 - 2 = 5x - 2x \]
\[ 18 = 3x \]
\[ x = \frac{18}{3} \]
\[ x = 6 \]
Step 4: Final Answer:
The value of \( x \) is 6.