<p>Let \(S = \dfrac{4}{19} + \dfrac{44}{19^2} + \dfrac{444}{19^3} + \cdots\) up to \(\infty\). Then \(S\) is equal to</p>
Step-by-Step Solution
Key Concept: Recognize that 4, 44, 444, ... can be written as 4(1, 11, 111, ...) where each numerator equals 4·(10^n - 1)/9. This transforms the series into a geometric series after algebraic manipulation.
<p><strong>Step 1:</strong> Express the numerators in a recognizable form.</p><p>Note that 4 = 4·1, 44 = 4·11, 444 = 4·111, etc.</p><p>Also, 1 = (10-1)/9, 11 = (10²-1)/9, 111 = (10³-1)/9, ...</p><p>So the numerator pattern is: 4·(10^n - 1)/9</p><p><strong>Step 2:</strong> Rewrite the series.</p><p>S = Σ[n=1 to ∞] [4(10^n - 1)/(9·19^n)]</p><p>S = (4/9)·Σ[n=1 to ∞] [(10^n - 1)/19^n]</p><p>S = (4/9)·[Σ(10/19)^n - Σ(1/19)^n]</p><p><strong>Step 3:</strong> Apply geometric series formula.</p><p>For |r| < 1: Σ[n=1 to ∞] r^n = r/(1-r)</p><p>Σ(10/19)^n = (10/19)/(1 - 10/19) = (10/19)/(9/19) = 10/9</p><p>Σ(1/19)^n = (1/19)/(1 - 1/19) = (1/19)/(18/19) = 1/18</p><p><strong>Step 4:</strong> Calculate S.</p><p>S = (4/9)·(10/9 - 1/18) = (4/9)·(20/18 - 1/18) = (4/9)·(19/18)</p><p>S = 76/162 = 38/81</p><p>∴ Answer: A</p>
Correct Answer: A