Sequences & Series
Arithmetic Progression - General Term
Grade 11
Question:
<p>If <strong>a</strong><sub>n</sub> = (x)<sup>1/(2n)</sup> + (y)<sup>1/(2n)</sup> and <strong>b</strong><sub>n</sub> = (x)<sup>1/(2n)</sup> - (y)<sup>1/(2n)</sup> for all n ∈ ℕ, then a<sub>1</sub>a<sub>2</sub>a<sub>3</sub>...a<sub>n</sub> is equal to</p>
<p>(a) \(x - y\)</p>
<p>(b) \(\frac{x + y}{b_n}\)</p>
<p>(c) \(\frac{x - y}{b_n}\)</p>
<p>(d) \(\frac{xy}{b_n}\)</p>
Step-by-Step Solution
Key Concept: Use the algebraic identity (a+b)(a-b) = a² - b² repeatedly to telescope the product. Each factor aₙ·bₙ = (x^(1/2ⁿ))² - (y^(1/2ⁿ))² = x^(1/2ⁿ⁻¹) - y^(1/2ⁿ⁻¹), creating a telescoping pattern.
<p><strong>Step 1: Write out the definitions clearly</strong></p><p>Given: aₙ = x^(1/2ⁿ) + y^(1/2ⁿ) and bₙ = x^(1/2ⁿ) - y^(1/2ⁿ)</p><p><strong>Step 2: Recognize the key relationship using difference of squares</strong></p><p>Consider the product aₙ·bₙ:</p><p>aₙ·bₙ = (x^(1/2ⁿ) + y^(1/2ⁿ))(x^(1/2ⁿ) - y^(1/2ⁿ)) = (x^(1/2ⁿ))² - (y^(1/2ⁿ))² = x^(1/2ⁿ⁻¹) - y^(1/2ⁿ⁻¹)</p><p><strong>Step 3: Apply this to build a telescoping product</strong></p><p>a₁·b₁ = x^(1/2) - y^(1/2)</p><p>a₂·b₂ = x^(1/4) - y^(1/4)</p><p>a₃·b₃ = x^(1/8) - y^(1/8)</p><p>And so on: aₙ·bₙ = x^(1/2ⁿ) - y^(1/2ⁿ)</p><p><strong>Step 4: Calculate a₁a₂a₃...aₙ using the telescoping pattern</strong></p><p>Multiply the telescoping equations strategically:</p><p>a₁·a₂·a₃·...·aₙ·(b₁·b₂·b₃·...·bₙ) = (x^(1/2) - y^(1/2))(x^(1/4) - y^(1/4))(x^(1/8) - y^(1/8))...(x^(1/2ⁿ) - y^(1/2ⁿ))</p><p>Notice: (x - y) = (x^(1/2) + y^(1/2))(x^(1/2) - y^(1/2)) = a₁·b₁</p><p>Continuing: (x - y) = a₁·b₁·a₂·b₂·a₃·b₃·...·aₙ·bₙ / (a₂·a₃·...·aₙ·b₂·b₃·...·bₙ)</p><p><strong>Step 5: Use the product telescoping property</strong></p><p>From (x^(1/2) - y^(1/2))(x^(1/2) + y^(1/2)) = x - y, we have:</p><p>a₁·b₁·a₂·b₂·...·aₙ·bₙ = (x - y)·(product of remaining terms)</p><p>More directly: (x - y) = a₁·b₁·a₂·b₂·a₃·b₃·...·aₙ·bₙ</p><p>Therefore: a₁·a₂·a₃·...·aₙ = (x - y)/(b₁·b₂·b₃·...·bₙ) = (x - y)/bₙ</p><p><strong>∴ Answer: c</strong></p>
Correct Answer: c