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

To determine the value of \( x \) for which the numbers \( 2x, (x + 10), \) and \( (3x + 2) \) form three consecutive terms of an Arithmetic Progression (A.P.), we need to understand the properties of an A.P.

In an A.P., the difference between consecutive terms is constant. Therefore, for the three terms to be in an A.P., the difference between the first and second terms should be equal to the difference between the second and third terms.

The terms given are:

• First term: \( a_1 = 2x \)
• Second term: \( a_2 = x + 10 \)
• Third term: \( a_3 = 3x + 2 \)

The conditions for an A.P. are:

\(a_2 - a_1 = a_3 - a_2\)

Substitute the values of \( a_1, a_2, \) and \( a_3 \):

\((x + 10) - 2x = (3x + 2) - (x + 10)\)

Simplify both sides of the equation:

Left side:

\(x + 10 - 2x = -x + 10\)

Right side:

\(3x + 2 - x - 10 = 2x - 8\)

Equating the simplified expressions:

\(-x + 10 = 2x - 8\)

Rearrange to get all terms involving \( x \) on one side:

\(10 + 8 = 2x + x\)

Simplifying further gives:

\(18 = 3x\)

Divide both sides by 3 to solve for \( x \):

\(x = \frac{18}{3} = 6\)

Therefore, the value of \( x \) is 6.

Conclusion: The correct answer is 6, which ensures that \( 2x, (x + 10), (3x + 2) \) form three consecutive terms of an A.P.