Question:

<p>The series can be written as:</p><p>$$1.4 + 3.04 + 5.004 + 7.0004 + \ldots$$</p><p>Separating integer and decimal parts:</p><p>$$(1 + 3 + 5 + 7 + \ldots) + (0.4 + 0.04 + 0.004 + 0.0004 + \ldots)$$</p><p>The first part is the sum of first n odd numbers = $n^2$</p><p>The second part is a geometric series: $$0.4 + 0.04 + 0.004 + \ldots = \frac{0.4}{1 - 0.1} = \frac{0.4}{0.9} = \frac{4}{9}$$</p><p>However, for n terms, the geometric part sums to $\frac{4}{9}\left(1 - 10^{-n}\right) = \frac{4}{9} - \frac{4}{9 \cdot 10^n}$</p><p>Total sum = $n^2 + \frac{4}{9}\left(1 - 10^{-n}\right) ≈ n^2 + \frac{n}{9}$ for practical purposes.</p><p>∴ Answer is (b)</p>