<p>The sum of series \(1 + \dfrac{4}{5} + \dfrac{7}{5^2} + \dfrac{10}{5^3} + \cdots\) to ∞ is</p>
Step-by-Step Solution
Key Concept: Recognize this as a sum of arithmetic progression numerators divided by geometric progression denominators. Use the technique of multiplying by the common ratio and subtracting to isolate the series.
<p><strong>Step 1:</strong> Identify the series structure. Numerators form AP: 1, 4, 7, 10, ... (first term a=1, common difference d=3). Denominators form GP with ratio r=1/5.</p><p><strong>Step 2:</strong> Let S = 1 + 4/5 + 7/5² + 10/5³ + ...</p><p>Multiply by r = 1/5: S/5 = 1/5 + 4/5² + 7/5³ + 10/5⁴ + ...</p><p><strong>Step 3:</strong> Subtract: S - S/5 = 1 + 3/5 + 3/5² + 3/5³ + ...</p><p>4S/5 = 1 + 3(1/5 + 1/5² + 1/5³ + ...)</p><p><strong>Step 4:</strong> The geometric series: 1/5 + 1/5² + ... = (1/5)/(1 - 1/5) = (1/5)/(4/5) = 1/4</p><p>4S/5 = 1 + 3(1/4) = 1 + 3/4 = 7/4</p><p><strong>Step 5:</strong> S = (7/4) × (5/4) = 35/16</p><p>∴ Answer: <strong>35/16</strong></p>
Correct Answer: D