Sequences & Series
Arithmetic and Geometric Progressions
Grade 11
Question:
<p>Let \(x, y\) are real numbers such that \(x, x + 2y, 2x + y\) form an A.P. while the numbers \((y + 1)^2, xy + 5, (x + 1)^2\) form a G.P., then \(|x| - |y|\) is equal to</p>
<p>(A) 0</p>
<p>(B) 1</p>
<p>(C) 2</p>
<p>(D) 4</p>
Step-by-Step Solution
Key Concept: Use the A.P. condition to express one variable in terms of the other, then use the G.P. condition to find specific values.
<p><strong>Step 1:</strong> From A.P. condition: \(x + 2y\) is the mean of \(x\) and \(2x + y\), so \(2(x + 2y) = x + 2x + y\), which gives \(3y = x\).</p><p><strong>Step 2:</strong> From G.P. condition: \((xy + 5)^2 = (y + 1)^2(x + 1)^2\).</p><p><strong>Step 3:</strong> Substitute \(x = 3y\) into the G.P. equation and solve for \(y\).</p><p><strong>Step 4:</strong> Calculate \(|x| - |y|\) to get 1.</p>
Correct Answer: B