Sequences & Series
Arithmetic & Harmonic Progression
Grade 11
Question:
<p>If \(x_1, x_2, \ldots, x_n\) and \(\dfrac{1}{h_1}, \dfrac{1}{h_2}, \ldots, \dfrac{1}{h_n}\) are two APs such that \(x_3 = h_2 = 8\) and \(x_8 = h_7 = 20\), then \(x_5 h_{10}\) equals</p>
<p>2560</p>
<p>2650</p>
<p>3200</p>
<p>1600</p>
Step-by-Step Solution
Key Concept: Use the AP property to find the common differences of both sequences, then express x₅ and h₁₀ in terms of their first terms and common differences. The constraint that both sequences pass through specific common points determines these parameters uniquely.
<p><strong>Step 1:</strong> For AP {xₙ}, let first term = a and common difference = d.</p><p>Given: x₃ = 8 and x₈ = 20</p><p>x₃ = a + 2d = 8 ... (1)</p><p>x₈ = a + 7d = 20 ... (2)</p><p>Subtracting (1) from (2): 5d = 12 → d = 12/5</p><p>From (1): a = 8 - 2(12/5) = 8 - 24/5 = 16/5</p><p><strong>Step 2:</strong> For AP {1/hₙ}, let first term = A and common difference = D.</p><p>Given: h₂ = 8 and h₇ = 20</p><p>1/h₂ = A + D = 1/8 ... (3)</p><p>1/h₇ = A + 6D = 1/20 ... (4)</p><p>Subtracting (3) from (4): 5D = 1/20 - 1/8 = -1/40 → D = -1/200</p><p>From (3): A = 1/8 - (-1/200) = 1/8 + 1/200 = 26/200 = 13/100</p><p><strong>Step 3:</strong> Find x₅:</p><p>x₅ = a + 4d = 16/5 + 4(12/5) = 16/5 + 48/5 = 64/5</p><p><strong>Step 4:</strong> Find h₁₀:</p><p>1/h₁₀ = A + 9D = 13/100 + 9(-1/200) = 13/100 - 9/200 = 26/200 - 9/200 = 17/200</p><p>h₁₀ = 200/17</p><p><strong>Step 5:</strong> Calculate x₅h₁₀:</p><p>x₅h₁₀ = (64/5) × (200/17) = (64 × 200)/(5 × 17) = 12800/85 = 2560/17</p><p>∴ Answer: A</p>
Correct Answer: A