<p>Let there be a G.P. whose first term is 'a' and common ratio is 'r'. If A and H are the arithmetic mean and the harmonic mean respectively for the first 'n' terms of the G.P. Then \(A \times H\) is equal to</p>
Step-by-Step Solution
Key Concept: For a G.P., the product of AM and HM of n terms equals the product of first and last terms.
<p><strong>Solution:</strong> For a G.P. with first term a and common ratio r, the n terms are: \(a, ar, ar^2, \ldots, ar^{n-1}\)</p><p>Arithmetic Mean: \(A = \frac{a + ar + ar^2 + \ldots + ar^{n-1}}{n} = \frac{a(r^n-1)}{n(r-1)}\)</p><p>For Harmonic Mean, we use: \(H = \frac{n}{\frac{1}{a} + \frac{1}{ar} + \ldots + \frac{1}{ar^{n-1}}}\)</p><p>This equals: \(H = \frac{n \cdot ar^{n-1}}{1 + r^{-1} + r^{-2} + \ldots + r^{-(n-1)}}\)</p><p>By the property that for G.P., \(A \times H = (\text{first term}) \times (\text{last term})\)</p><p>\(A \times H = a \times ar^{n-1} = a^2r^{n-1}\)</p><p>∴ Answer is B.</p>
Correct Answer: B