Question

(2-1/3)(2-3/5)(2-5/7)... ... ... ..(2-999/1001)=

Answer 1

$\displaystyle \left(2-\frac{1}{3}\right)\left(2-\frac{3}{5}\right)\left(2-\frac{5}{7}\right)\left(2-\frac{7}{9}\right)\dots\dots\left(2-\frac{999}{1001}\right) \\ \\ \\ \left(\frac{6-1}{3}\right)\left(\frac{10-3}{5}\right)\left(\frac{14-5}{7}\right)\left(\frac{18-7}{9}\right)\dots\dots\left(\frac{2002-999}{1001}\right) \\ \\ \\ \frac{5}{3}\cdot\frac{7}{5}\cdot\frac{9}{7}\cdot\frac{11}{9}\cdot\frac{13}{11} \cdot \dots \dots \ \cdot\frac{1001}{999}\cdot\frac{1003}{1001} \\ \\ \\ \frac{1003}{3}$

$\displaystyle \Large \text{ \bold{334\ \frac{1}{3}} }$

Thus we got the answer!

First we have to subtract each fraction in each brackets from 2.

After subtraction, we found out that the same numerator of one fraction is appeared as the denominator of the next fraction, like,

→  the numerator of the first fraction 5 is appeared as denominator of the second fraction.

→  the numerator of the second fraction 7 is appeared as denominator of the third fraction.

→  the numerator of the third fraction 5 is appeared as denominator of the fourth fraction.

etc.

So these same numerators and denominators would be cut off, and 1003 and 3 would remain as numerator and denominator respectively.

Thus the answer would be 1003/3, or 334  1/3.