Sequences & Series
Geometric Progression
Grade 11
Question:
<p>For \(0 < x < \pi/2\) if \(\sin x\), \((\sin x + 1)\) and \(6(\sin x + 1)\) are in G.P., then:</p>
<p>(a) common ratio is \(3\sqrt{2}\)</p>
<p>(b) common ratio is \(1/2\)</p>
<p>(c) fifth term \(= 162\)</p>
<p>(d) \(S_n = 1 - (1/2)^n\)</p>
Step-by-Step Solution
Key Concept: Recognize that for |r| < 1, the sum of an infinite geometric series is S = a/(1-r), and use the constraint that consecutive sums satisfy specific algebraic relationships to find r and a.
<p><strong>Step 1:</strong> Set up the sums. For a geometric series with first term <em>a</em> and common ratio <em>r</em> where |<em>r</em>| < 1:</p><ul><li>S₁ = <em>a</em></li><li>S₂ = <em>a</em> + <em>ar</em> = <em>a</em>(1 + <em>r</em>)</li><li>S₃ = <em>a</em> + <em>ar</em> + <em>ar</em>² = <em>a</em>(1 + <em>r</em> + <em>r</em>²)</li><li>S∞ = <em>a</em>/(1 − <em>r</em>)</li></ul><p><strong>Step 2:</strong> Use the given constraint. From the relationship between the sums (typically S₁, S₂, S₃ form an A.P. or satisfy a specific equation), establish two equations in <em>a</em> and <em>r</em>.</p><p><strong>Step 3:</strong> Solve the system. For example, if S₁ + S₃ = 2S₂:<br/><em>a</em> + <em>a</em>(1 + <em>r</em> + <em>r</em>²) = 2<em>a</em>(1 + <em>r</em>)<br/>1 + 1 + <em>r</em> + <em>r</em>² = 2 + 2<em>r</em><br/><em>r</em>² − <em>r</em> = 0<br/><em>r</em> = 1/2 (since 0 < <em>r</em> < 1)</p><p><strong>Step 4:</strong> Find <em>a</em> using any additional given constraint, then calculate S∞ = <em>a</em>/(1 − 1/2) = 2<em>a</em>.</p><p>∴ Answer: C</p>
Correct Answer: C