Find the sum of : \[ 1 + 0.6 + 0.06 + 0.006 + 0.0006 + \dots \]

Practice MCQ for SSC, UPSC, RRB Exams

A. \(1\frac{2}{3}\) A. \(1\frac{2}{3}\)

B. \(1\frac{1}{3}\) B. \(1\frac{1}{3}\)

C. \(2\frac{1}{3}\) C. \(2\frac{1}{3}\)

D. \(2\frac{2}{3}\) D. \(2\frac{2}{3}\)

The given series is:

\[ 1 + 0.6 + 0.06 + 0.006 + \dots \]

This is a Geometric Progression (G.P.)

First term \(a = 1\) Common ratio \(r = 0.6\)

Since \(|r| < 1\), sum to infinity:

\[ S = \frac{a}{1 - r} \] \[ = \frac{1}{1 - 0.6} \] \[ = \frac{1}{0.4} \] \[ = 2.5 \] \[ = 2\frac{1}{2} \]

Since \(2\frac{1}{2}\) is not in the options, observe carefully:

Actually the terms after \(0.6\) are \(0.06, 0.006,\dots\) which form another G.P. with:

\[ a = 0.6, \quad r = 0.1 \]

Sum of that part:

\[ = \frac{0.6}{1 - 0.1} = \frac{0.6}{0.9} = \frac{2}{3} \]

Total sum:

\[ 1 + \frac{2}{3} = \frac{5}{3} = 1\frac{2}{3} \]

Hence, the correct answer is \(1\frac{2}{3}\).

ধারাটি হলো:

\[ 1 + 0.6 + 0.06 + 0.006 + \dots \]

এটি একটি জ্যামিতিক ধারা।

প্রথম পদ \(a = 1\) অনুপাত \(r = 0.6\)

কিন্তু লক্ষ্য করলে দেখা যায়, \(0.6, 0.06, 0.006,\dots\) আলাদা একটি জ্যামিতিক ধারা:

\[ a = 0.6, \quad r = 0.1 \]

এই অংশের যোগফল:

\[ = \frac{0.6}{1 - 0.1} = \frac{0.6}{0.9} = \frac{2}{3} \]

অতএব মোট যোগফল:

\[ 1 + \frac{2}{3} = \frac{5}{3} = 1\frac{2}{3} \]

সুতরাং সঠিক উত্তর \(1\frac{2}{3}\)।

Find the sum of : \[ 1 + 0.6 + 0.06 + 0.006 + 0.0006 + \dots \] নিচের ধারাটির যোগফল নির্ণয় করো: \[ 1 + 0.6 + 0.06 + 0.006 + \dots \]