Sequences & Series
Geometric Progression
Grade 11
Question:
<p>If <i>a</i>, <i>b</i>, <i>c</i> are <i>p</i>th, <i>q</i>th and <i>r</i>th terms of a GP, then <i>(q - r)</i> log <i>a</i> + (<i>r - p</i>) log <i>b</i> + (<i>p - q</i>) log <i>c</i> is equal to</p>
<p>(a) <i>p + q + r</i></p>
<p>(b) 1</p>
<p>(c) <i>-pqr</i></p>
<p>(d) 0</p>
Step-by-Step Solution
Key Concept: Express each term of the GP in terms of first term and common ratio, then use logarithm properties to simplify the expression.
<p><strong>Solution:</strong></p><p>Let <i>A</i> and <i>R</i> be the first term and common ratio of the given GP.</p><p>Then, <i>a = AR</i><sup><i>p-1</i></sup></p><p>∴ log <i>a</i> = log <i>A</i> + (<i>p</i> - 1) log <i>R</i> ... (i)</p><p>Similarly, log <i>b</i> = log <i>A</i> + (<i>q</i> - 1) log <i>R</i> ... (ii)</p><p>and log <i>c</i> = log <i>A</i> + (<i>r</i> - 1) log <i>R</i> ... (iii)</p><p>Now, (<i>q - r</i>) log <i>a</i> + (<i>r - p</i>) log <i>b</i> + (<i>p - q</i>) log <i>c</i></p><p>= (<i>q - r</i>){log <i>A</i> + (<i>p</i> - 1) log <i>R</i>} + (<i>r - p</i>){log <i>A</i> + (<i>q</i> - 1) log <i>R</i>} + (<i>p - q</i>){log <i>A</i> + (<i>r</i> - 1) log <i>R</i>}</p><p>= log <i>A</i>[<i>q - r + r - p + p - q</i>] + log <i>R</i>[<i>p</i>(<i>q - r</i>) + <i>q</i>(<i>r - p</i>) + <i>r</i>(<i>p - q</i>) - (<i>q - r</i>) - (<i>r - p</i>) - (<i>p - q</i>)]</p><p>= log <i>A</i> × 0 + log <i>R</i> × 0 = 0</p><p>∴ Answer is (d) 0.</p>
Correct Answer: D