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?
Understanding Arithmetic Progression and Logarithms
The question asks us to determine the validity of two statements based on a condition involving terms in an Arithmetic Progression (AP).
Condition from Arithmetic Progression
We are given that the terms \((10+\log_{10} x)\), \((10+\log_{10} y)\), and \((10+\log_{10} z)\) are in Arithmetic Progression.
For any three terms \(a\), \(b\), and \(c\) to be in AP, the middle term is the average of the other two, meaning \(2b = a + c\). Applying this rule to our terms:
\(2(10+\log_{10} y) = (10+\log_{10} x) + (10+\log_{10} z)\)
Let's simplify this equation:
\(20 + 2\log_{10} y = 10 + \log_{10} x + 10 + \log_{10} z\)
\(20 + 2\log_{10} y = 20 + \log_{10} x + \log_{10} z\)
Subtracting 20 from both sides gives:
\(2\log_{10} y = \log_{10} x + \log_{10} z\)
Using the logarithm property \(n \log a = \log a^n\), we can rewrite the left side:
\(\log_{10} (y^2) = \log_{10} x + \log_{10} z\)
Using the logarithm property \(\log a + \log b = \log (ab)\), we can rewrite the right side:
\(\log_{10} (y^2) = \log_{10} (xz)\)
Since the logarithms are equal and the base is the same, their arguments must be equal:
\(y^2 = xz\)
Analysis of Statement I
Statement I says: "The GM of \(x\) and \(z\) is \(y^2\)."
The Geometric Mean (GM) of two numbers \(a\) and \(b\) is defined as \(\sqrt{ab}\).
So, the GM of \(x\) and \(z\) is \(\sqrt{xz}\).
From our AP calculation, we found that \(xz = y^2\).
Therefore, the GM of \(x\) and \(z\) is \(\sqrt{y^2}\). Assuming \(y\) is positive (as \(\log_{10} y\) is involved), \(\sqrt{y^2} = y\).
The statement claims the GM is \(y^2\), but we found it to be \(y\). Thus, Statement I is incorrect.
Analysis of Statement II
Statement II says: "The AM of \(\log_{10} x\) and \(\log_{10} z\) is \(\log_{10} y\)."
The Arithmetic Mean (AM) of two numbers \(a\) and \(b\) is defined as \(\frac{a+b}{2}\).
So, the AM of \(\log_{10} x\) and \(\log_{10} z\) is:
\(AM = \frac{\log_{10} x + \log_{10} z}{2}\)
From our simplification of the AP condition, we derived:
\(2\log_{10} y = \log_{10} x + \log_{10} z\)
Dividing both sides by 2 gives:
\(\frac{2\log_{10} y}{2} = \frac{\log_{10} x + \log_{10} z}{2}\)
\(\log_{10} y = \frac{\log_{10} x + \log_{10} z}{2}\)
This shows that the AM of \(\log_{10} x\) and \(\log_{10} z\) is indeed equal to \(\log_{10} y\). Thus, Statement II is correct.
Conclusion
Based on the analysis:
- Statement I is incorrect.
- Statement II is correct.
Therefore, only Statement II is correct.
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The sum of the first three terms of an arithmetic progression (A.P.) is $24$, and the sum of its next three terms (i.e., the $4^{th}$, $5^{th}$, and $6^{th}$ terms) is $51$. What is the sum of the first $10$ terms of this A.P.?
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The sequence given by interval [0,1] is ______.
Find the limit point of the sequence <1, 2, 1/2, 3, 1/3..... >
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