Institution: বঙ্গবন্ধু শেখ মুজিবুর রহমান বিজ্ঞান ও প্রযুক্তি বিশ্ববিদ্যালয় (BSMRSTU) Exam: A ইউনিট || বঙ্গবন্ধু শেখ মুজিবুর রহমান বিজ্ঞান ও প্রযুক্তি বিশ্ববিদ্যালয় || 2011
Topic: Physics (পদার্থবিদ্যা), Subject: General Science (সাধারণ বিজ্ঞান)

Best Answer Explained in Below-:

The series \(1 + 0.2 + 0.04 + 0.0008 + \ldots\) can be identified as a geometric series. In a geometric series, each term after the first is obtained by multiplying the previous term by a constant ratio.

Here, we can express the terms as follows:
- The first term \(a = 1\),
- The second term \(0.2\) can be written as \(1 \cdot 0.2\),
- The third term \(0.04\) is \(0.2 \cdot 0.2\) or \(1 \cdot (0.2)^2\),
- This continues with \(0.0008 = 1 \cdot (0.2)^3\), and so on.

The general term of this series can then be written as \(1 \cdot (0.2)^n\) for \(n = 0, 1, 2, \ldots\).

The common ratio \(r\) in this geometric series is \(0.2\). The sum \(S\) of an infinite geometric series can be calculated using the formula:

\[
S = \frac{a}{1 - r}
\]

where \(a\) is the first term and \(r\) is the common ratio. Here, substituting \(a = 1\) and \(r = 0.2\):

\[
S = \frac{1}{1 - 0.2} = \frac{1}{0.8} = \frac{5}{4}
\]

Thus, the sum of the series \(1 + 0.2 + 0.04 + 0.0008 + \ldots\) converges to \(\frac{5}{4}\), confirming that the correct answer is option A: \(\frac{5}{4}\).