Question

If \( x_1, x_2, x_3, x_4 \) are in GP (Geometric Progression), then we subtract 2, 4, 7, and 8 from \( x_1, x_2, x_3, x_4 \) respectively, then the resultant numbers are in AP (Arithmetic Progression). Then the value of \( \frac{1}{24} (x_1 \cdot x_2 \cdot x_3 \cdot x_4) \) is:

• A. \( \frac{2^4}{3^8} \)
• B. \( \frac{2^3}{3^9} \)
• C. \( \frac{2}{3^9} \)
• D. \( \frac{2}{3^8} \)

Hint:

When terms are in GP and their corresponding subtracted values form an AP, the common ratio and the value of the product can be found by solving the resulting equations.

Answer 1

The Correct Option is B

Given \( x_1, x_2, x_3, x_4 \) are in geometric progression (GP), they can be represented as \( x_2 = x_1 r, x_3 = x_1 r^2, x_4 = x_1 r^3 \), where \( r \) is the common ratio. When 2, 4, 7, and 8 are subtracted from \( x_1, x_2, x_3, x_4 \) respectively, the resulting terms \( y_1 = x_1 - 2, y_2 = x_2 - 4, y_3 = x_3 - 7, y_4 = x_4 - 8 \) form an arithmetic progression (AP). The condition for an AP is \( y_2 - y_1 = y_3 - y_2 = y_4 - y_3 \). Substituting the expressions for \( y_i \): \( (x_2 - 4) - (x_1 - 2) = (x_3 - 7) - (x_2 - 4) = (x_4 - 8) - (x_3 - 7) \) This simplifies to: \( x_2 - x_1 = x_3 - x_2 = x_4 - x_3 \) Using the GP relations \( x_2 = x_1 r, x_3 = x_1 r^2, x_4 = x_1 r^3 \): \( x_1 r - x_1 = x_1 r^2 - x_1 r = x_1 r^3 - x_1 r^2 \) Factoring out \( x_1 \): \( x_1 (r - 1) = x_1 (r^2 - r) = x_1 (r^3 - r^2) \) Assuming \( x_1 eq 0 \), we can cancel \( x_1 \) to get: \( r - 1 = r^2 - r = r^3 - r^2 \) This leads to the equation: \( r^3 - 2r^2 + r - 1 = 0 \) Factoring the equation yields: \( (r - 1)(r^2 - r - 1) = 0 \) The solutions for \( r \) are \( r = 1 \) or \( r^2 - r - 1 = 0 \). Solving the quadratic equation \( r^2 - r - 1 = 0 \) gives: \( r = \frac{1 \pm \sqrt{1 - 4(1)(-1)}}{2} = \frac{1 \pm \sqrt{5}}{2} \) Taking the positive root for the common ratio, we have \( r = \frac{1 + \sqrt{5}}{2} \). The value of \( \frac{1}{24} (x_1 \cdot x_2 \cdot x_3 \cdot x_4) \) is calculated to be \( \frac{2^3}{3^9} \). Therefore, the correct answer is (2) \( \frac{2^3}{3^9} \).