• # question_answer Read the following statements carefully and choose the correct option. Value of $\left( 1-\frac{1}{2} \right)\,\left( 1-\frac{1}{3} \right)\,\left( 1-\frac{1}{4} \right).....\left( 1-\frac{1}{x} \right)$ is equal to $\frac{1}{x}$. (ii) If $2=x+\frac{1}{1+\frac{1}{3+\frac{1}{4}}}$ then value of x is $\frac{21}{17}$ (iii) Value of $999\frac{1}{7}+999\frac{2}{7}\,+999\frac{3}{7}+999\frac{4}{7}$$+999\frac{5}{7}\,+999\frac{6}{7}$ is 5999. A) and (ii) are true while (iii) is false. B) and (iii) are true while (ii) is false. C) (ii) and (iii) are true while (i) is false. D) Neither (i) and (ii) nor (iii) is true.

(i) $\left( 1-\frac{1}{2} \right)\,\left( 1-\frac{1}{3} \right)\,\left( 1-\frac{1}{4} \right)\,.........\left( 1-\frac{1}{x} \right)$ $=\frac{1}{2}\times \frac{2}{3}\times \frac{3}{4}\times \,.........\times \frac{x-1}{x}=\frac{1}{x}$ (ii) $2=x+\frac{1}{1+\frac{1}{\frac{12+1}{4}}}\,\,\,\,\,\,\,\,\,=x+\frac{1}{1+\frac{1}{\frac{13}{4}}}$ $=x+\frac{1}{\frac{13+4}{13}}=x+\frac{13}{17}$ $\therefore \,\,\,\,\,\,\,\,\,\,x=12-\frac{13}{17}=\frac{21}{17}$ (iii) $(999\times 6)\,+\left( \frac{1}{7}+\frac{2}{7}+\frac{3}{7}+\frac{4}{7}+\frac{5}{7}+\frac{6}{7} \right)$ $=5994+\frac{21}{7}\,=5994+3=5997$