If (10+\log_{10}x), (10+\log_{10}y) and (10+\log_{10}z) are in AP, then consider the following statements: I. The GM of x and z is y^{2}. II. The AM of \log_{10}x and \log_{10}z is \log_{10}y. Which of the statements given above is/are correct?

Explanation

Since the terms are in AP, 2(10 + \log_{10}y) = (10 + \log_{10}x) + (10 + \log_{10}z). This simplifies to 2\log_{10}y = \log_{10}x + \log_{10}z, making \log_{10}y the AM of \log_{10}x and \log_{10}z (Statement II is true). This also means \log_{10}y^2 = \log_{10}(xz), so y^2 = xz. The GM of x and z is \sqrt{xz} = y, not y^2. Hence, Statement I is false.

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