Sequences & Series
Arithmetic Progression
Grade 11
Question:
<p>It is given that the numbers \(3, 3\log_y x, 3\log_z y, 7\log_x z\) form an arithmetic progression. Then</p>
<p>(A) \(x^6 = y^8\)</p>
<p>(B) \(y^3 = z^4\)</p>
<p>(C) \(x^6 = y^7\)</p>
<p>(D) \(x^9 = z^{14}\)</p>
Step-by-Step Solution
Key Concept: Convert logarithms to a common base and use the A.P. condition to establish relationships between the variables.
<p>For the numbers to be in A.P., the common difference must be constant. Let the common difference be \(d\). Then:\n\(3\log_y x - 3 = d\)\n\(3\log_z y - 3\log_y x = d\)\n\(7\log_x z - 3\log_z y = d\)\nUsing logarithm properties and converting to a common base, we get \(\log x / \log y = 3/2\) and \(\log y / \log z = 4/3\). This yields \(y^3 = z^4\).</p>
Correct Answer: B