Sequences & Series
Harmonic Progression
Grade 11
Question:
<p>If in a progression \(a_1, a_2, a_3, \ldots\) etc., \((a_r - a_{r+1})\) bears a constant ratio with \(a_r \times a_{r+1}\), then the terms of the progression are in</p>
<p>A.P.</p>
<p>G.P.</p>
<p>H.P.</p>
<p>None of these</p>
Step-by-Step Solution
Key Concept: If (aᵣ - aᵣ₊₁) is proportional to aᵣ × aᵣ₊₁, then (aᵣ - aᵣ₊₁)/(aᵣ × aᵣ₊₁) = constant. Dividing both numerator and denominator by aᵣ × aᵣ₊₁ yields 1/aᵣ₊₁ - 1/aᵣ = constant, revealing the reciprocals form an AP.
<p><strong>Step 1:</strong> Given that (aᵣ - aᵣ₊₁) bears a constant ratio with aᵣ × aᵣ₊₁, we write:</p><p>(aᵣ - aᵣ₊₁) = k(aᵣ × aᵣ₊₁), where k is a constant</p><p><strong>Step 2:</strong> Divide both sides by aᵣ × aᵣ₊₁:</p><p>(aᵣ - aᵣ₊₁)/(aᵣ × aᵣ₊₁) = k</p><p><strong>Step 3:</strong> Separate the fraction on the left side:</p><p>aᵣ/(aᵣ × aᵣ₊₁) - aᵣ₊₁/(aᵣ × aᵣ₊₁) = k</p><p>1/aᵣ₊₁ - 1/aᵣ = k</p><p><strong>Step 4:</strong> Rearrange:</p><p>1/aᵣ - 1/aᵣ₊₁ = -k</p><p>This shows that consecutive terms of the sequence {1/aᵣ} have a constant difference of -k.</p><p><strong>Step 5:</strong> By definition, a sequence where consecutive terms have a constant common difference is in Arithmetic Progression (AP).</p><p>∴ <strong>Answer: C</strong> — The terms are in <strong>Harmonic Progression (HP)</strong>, since their reciprocals form an AP.</p>
Correct Answer: C