• # question_answer $2.\overset{\bullet \,\,\bullet \,\,\bullet }{\mathop{357}}\,=$ [IIT 1983; RPET 1995] A) $\frac{2355}{1001}$ B) $\frac{2370}{997}$ C) $\frac{2355}{999}$ D) None of these

Given that $2.\overset{.\,\,\,.\,\,\,.}{\mathop{357}}\,=2.357357357357......$ $=2+0.\overset{{}}{\mathop{357}}\,+0.000357+0.000000357+.......\infty$ $=2+\frac{357}{{{10}^{3}}}+\frac{357}{{{10}^{6}}}+\frac{357}{{{10}^{9}}}+.......$ $\therefore$ ${{S}_{\infty }}=2+\frac{\frac{357}{{{10}^{3}}}}{1-\frac{1}{{{10}^{3}}}}=2+\frac{357}{{{10}^{3}}}\times \frac{{{10}^{3}}}{999}=\frac{2355}{999}$.