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    Question

    In this problem, two quantities, I and II, are provided. You are required to solve both quantities and determine the appropriate relationship between Quantity-I and Quantity-II. 

    Three positive numbers are in an

    arithmetic progression. The sum of these three numbers equals 108, and the sum of the squares of the first and second terms is 1872. Quantity-I: Determine the 69th term of the progression. Quantity-II: Calculate the product of the first two terms of the progression.
    A Quantity-I < Quantity-II Correct Answer Incorrect Answer
    B Quantity I > Quantity II Correct Answer Incorrect Answer
    C Quantity I = Quantity II or No relation can be established Correct Answer Incorrect Answer
    D Quantity-I ≤ Quantity-II Correct Answer Incorrect Answer
    E Quantity I ≥ Quantity II Correct Answer Incorrect Answer

    Solution

    ATQ, Let the first term be 'a' and common difference be 'd' ATQ; a + (a + d) + (a + 2d) = 108 or, 3a + 3d = 108 so, a + d = 36 .......... (I) ATQ; (a + d)2 + a2 = 1872 Or, 362 + a2 = 1872 Or, a2 = 576 So, a = ±24 (Since, the given three numbers are positive, a = 24) So, d = 36 - 24 = 12 Quantity I: 69th  term = a + (69 - 1) × d = a + 68d = 24 + 68 × 12 = 840 So, Quantity I = 840 Quantity II: First term = 24 Second term = 24 + 12 = 36 So, required product = 24 × 36 = 864 So, Quantity II = 864 So, Quantity II > Quantity I

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