<p>Let the sum of first 5 terms of a G.P. be \(\dfrac{a}{r^2} + \dfrac{a}{r} + a + ar + ar^2\). If this sum equals 49 times the sum of corresponding reciprocals, and \(a\) is the middle term, find the value of \(a\).</p>
Step-by-Step Solution
Key Concept: Recognize that the given expression is a G.P. with first term a/r² and common ratio r, then use the symmetry property that for a G.P., the sum of terms equals a constant multiple of the sum of their reciprocals when properly related.
<p><strong>Step 1:</strong> Identify the G.P. structure. The sum S = a/r² + a/r + a + ar + ar² is a G.P. with 5 terms, first term A = a/r², and common ratio = r.</p><p><strong>Step 2:</strong> Find sum of reciprocals. Reciprocals are r²/a, r/a, 1/a, 1/(ar), 1/(ar²). This is also a G.P. with first term r²/a and common ratio 1/r. Their sum S' = (r²/a)[1 - (1/r)⁵]/[1 - 1/r] = (r²/a) · (r⁵ - 1)/(r⁴(r-1)).</p><p><strong>Step 3:</strong> Use the given condition. S = 49·S'. Note that the middle term a of the original G.P. is the 3rd term.</p><p><strong>Step 4:</strong> Simplify using symmetry. S/S' = 49 gives: [a/r² + a/r + a + ar + ar²]/[r²/a + r/a + 1/a + 1/(ar) + 1/(ar²)] = 49. By factoring: a(1/r² + 1/r + 1 + r + r²)/(1/a)(r² + r + 1 + 1/r + 1/r²) = 49.</p><p><strong>Step 5:</strong> Since the numerator and denominator have the same terms, we get: a²/1 = 49, so a² = 49.</p><p><strong>Step 6:</strong> Therefore, a = 7 (taking positive value for standard G.P. context).</p><p>∴ Answer: C</p>
Correct Answer: C