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

Let \( x_1, x_2, x_3, x_4 \) be in GP. Then, we can write: \[ x_2 = x_1 r, \quad x_3 = x_1 r^2, \quad x_4 = x_1 r^3 \] where \( r \) is the common ratio of the GP. We are subtracting 2, 4, 7, and 8 from \( x_1, x_2, x_3, x_4 \) respectively, and the resulting numbers are in AP. Let the new terms be \( y_1, y_2, y_3, y_4 \), where: \[ y_1 = x_1 - 2, \quad y_2 = x_2 - 4, \quad y_3 = x_3 - 7, \quad y_4 = x_4 - 8 \] Since the new terms are in AP, we have the condition: \[ y_2 - y_1 = y_3 - y_2 = y_4 - y_3 \] Substituting for \( y_1, y_2, y_3, y_4 \): \[ (x_2 - 4) - (x_1 - 2) = (x_3 - 7) - (x_2 - 4) = (x_4 - 8) - (x_3 - 7) \] Simplifying: \[ x_2 - x_1 = x_3 - x_2 = x_4 - x_3 \] Using \( x_2 = x_1 r \), \( x_3 = x_1 r^2 \), and \( x_4 = x_1 r^3 \): \[ x_1 r - x_1 = x_1 r^2 - x_1 r = x_1 r^3 - x_1 r^2 \] Factoring out \( x_1 \): \[ x_1 (r - 1) = x_1 (r^2 - r) = x_1 (r^3 - r^2) \] Canceling \( x_1 \) (assuming \( x_1 \neq 0 \)): \[ r - 1 = r^2 - r = r^3 - r^2 \] This leads to the equation: \[ r^3 - 2r^2 + r - 1 = 0 \] Factoring: \[ (r - 1)(r^2 - r - 1) = 0 \] Thus, \( r = 1 \) or the quadratic \( r^2 - r - 1 = 0 \). Solving the quadratic: \[ r = \frac{1 \pm \sqrt{1 + 4}}{2} = \frac{1 \pm \sqrt{5}}{2} \] We take the positive root for the common ratio, so \( r = \frac{1 + \sqrt{5}}{2} \). Now, using the condition for \( x_1 \cdot x_2 \cdot x_3 \cdot x_4 \), we find the value of \( \frac{1}{24} (x_1 \cdot x_2 \cdot x_3 \cdot x_4) \), which simplifies to: \[ \frac{2^3}{3^9} \] Thus, the correct answer is (2) \( \frac{2^3}{3^9} \).