Sequences & Series
AM and GM
Grade 11
Question:
<p>Let <i>G</i> be the geometric mean of two positive numbers <i>a</i> and <i>b</i>, and <i>M</i> be the arithmetic mean of \(\dfrac{1}{a}\) and \(\dfrac{1}{b}\). If \(\dfrac{1}{G} : M = 4 : 5\), then \(a : b\) can be</p>
<p>\(1 : 4\)</p>
<p>\(1 : 2\)</p>
<p>\(2 : 3\)</p>
<p>\(3 : 4\)</p>
Step-by-Step Solution
Key Concept: Express G and M in terms of a and b, then use the given ratio to establish a relationship. The geometric mean of a and b is √(ab), and the arithmetic mean of 1/a and 1/b is (a+b)/(2ab).
<p><strong>Step 1:</strong> Identify the means.</p><p>Geometric mean: G = √(ab)</p><p>Arithmetic mean of 1/a and 1/b: M = (1/a + 1/b)/2 = (a+b)/(2ab)</p><p><strong>Step 2:</strong> Express 1/G.</p><p>1/G = 1/√(ab)</p><p><strong>Step 3:</strong> Apply the given ratio.</p><p>1/G : M = 4 : 5</p><p>Therefore: (1/G)/M = 4/5</p><p>[1/√(ab)] / [(a+b)/(2ab)] = 4/5</p><p><strong>Step 4:</strong> Simplify.</p><p>[1/√(ab)] × [2ab/(a+b)] = 4/5</p><p>2√(ab)/(a+b) = 4/5</p><p>10√(ab) = 4(a+b)</p><p>100ab = 16(a+b)²</p><p><strong>Step 5:</strong> Let a/b = t, and divide by b².</p><p>100t = 16(t+1)²</p><p>100t = 16t² + 32t + 16</p><p>16t² - 68t + 16 = 0</p><p>4t² - 17t + 4 = 0</p><p><strong>Step 6:</strong> Solve using the quadratic formula.</p><p>t = [17 ± √(289-64)]/8 = [17 ± √225]/8 = [17 ± 15]/8</p><p>t = 4 or t = 1/4</p><p>∴ a : b = 4 : 1 or 1 : 4</p>
Correct Answer: A