Sequences & Series
Geometric Progression
Grade 11
Question:
<p>If \((p+q)\)th term of a G.P. is \('a'\) and its \((p-q)\)th term is \('b'\) where \(a, b > 0\), then find its \(p\)th term.</p>
<p>\(\sqrt{ab}\)</p>
<p>\(ab\)</p>
<p>\(\dfrac{a+b}{2}\)</p>
<p>\(\dfrac{a}{b}\)</p>
Step-by-Step Solution
Key Concept: In a G.P., the product of terms equidistant from a central term equals the square of that central term. Use T_(p+q) · T_(p-q) = (T_p)² to find the pth term.
<p><strong>Step 1:</strong> Let the first term be A and common ratio be r. Write the general term formula.</p><p>T_(p+q) = A·r^(p+q-1) = a ... (1)</p><p>T_(p-q) = A·r^(p-q-1) = b ... (2)</p><p><strong>Step 2:</strong> Multiply equations (1) and (2):</p><p>T_(p+q) · T_(p-q) = A·r^(p+q-1) · A·r^(p-q-1) = A²·r^(2p-2)</p><p>ab = A²·r^(2p-2) = (A·r^(p-1))²</p><p><strong>Step 3:</strong> Recognize that A·r^(p-1) = T_p (the pth term).</p><p>ab = (T_p)²</p><p><strong>Step 4:</strong> Since a, b > 0, we have T_p > 0:</p><p>T_p = √(ab)</p><p>∴ Answer: A (which is √(ab))</p>
Correct Answer: A