Consider a, b, c in a G.P. such that |a + b + c| = 15. The median of these…

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Consider a, b, c in a G.P. such that |a + b + c| = 15. The median of these three terms is a, and b = 10. If a > c, what is the product of the first 4 terms of this G.P.?

  1. A.

    40000

  2. B.

    32000

  3. C.

    8000

  4. D.

    48000

Attempted by 3 students.

Show answer & explanation

Correct answer: A

Step-by-Step Solution

  1. Determine the values of a and c: Since b = 10, the terms are a, 10, c. Because they are in a G.P., a * c = 10^2 = 100. We can express the terms as 10 / r, 10, 10 * r, where r is the common ratio. Here, a = 10 / r and c = 10 * r. We are given the sum of the absolute value of the terms as 15. Testing the case where a + b + c = -15: (10 / r) + 10 + 10 * r = -15 (10 / r) + 10 * r = -25 10 * (1 + r^2) = -25 * r 2 * (1 + r^2) = -5 * r 2 * r^2 + 5 * r + 2 = 0 (2 * r + 1) * (r + 2) = 0 This gives two possible ratios: r = -0.5 or r = -2.

  2. Evaluate the conditions:

    • If r = -2: the terms are 10 / -2, 10, 10 * -2, which are -5, 10, -20. The median of -5, 10, and -20 is -5. This matches our value for a. The condition a > c means -5 > -20, which is satisfied.

    • If r = -0.5: the terms are 10 / -0.5, 10, 10 * -0.5, which are -20, 10, -5. The median is -5. This matches a. Condition a > c means -5 > -20, which is satisfied.

  3. Calculate the product of the first 4 terms: Using the sequence -5, 10, -20, 40: Product = (-5) * 10 * (-20) * 40 = 40,000.

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