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    Question

    If a, b, c are in GP, then which of the following is

    always true ?
    A loga+logc=2logb Correct Answer Incorrect Answer
    B a+b+c=0 Correct Answer Incorrect Answer
    C b²=a+c Correct Answer Incorrect Answer
    D a = b + c Correct Answer Incorrect Answer

    Solution

    Given: a, b, c are in Geometric Progression (G.P.) In G.P., the basic identity is: b² = a × c We now check each option: (A) log a + log c = 2 log b

    • Left side: log a + log c = log(ac)
    • Right side: 2 log b = log(b ² )
    So: log(ac) = log(b²) ⇒ ac = b² This is always true in G.P. (B) a + b + c = 0 Not always true in G.P. Example: a = 1, b = 2, c = 4 → sum = 7 So, not always true (C) b² = a + c This is not a property of G.P. In G.P., b ² = ac And this is not a property of A.P. either — in A.P., we have 2b = a + c
    So this is not valid in general (D) a = b + c Also not true in G.P. in general. No standard identity supports this relation Therefore, the correct answer is option (A).

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