c = (1-1/1 2)(1-1/2 2 3)(1-1/1 2 3 4)...(1-1/1 2 3 ... 2016)

Lời giải:

\[\begin{array}{l}P = \left( {1 - \frac{1}{{1 + 2}}} \right)\left( {1 - \frac{1}{{1 + 2 + 3}}} \right)...\left( {1 - \frac{1}{{1 + 2 + 3 + ... + 2014}}} \right)\\ = \frac{{(1 + 2)2:2 - 1}}{{(1 + 2)2:2}}.\frac{{(1 + 3)3:2 - 1}}{{(1 + 3)3:2}}...\frac{{(1 + 2014)2014:2 - 1}}{{(1 + 2014)2014:2}}\\ = \frac{2}{{2.3:2}}.\frac{5}{{3.4:2}}....\frac{{2029104}}{{2014.2015:2}}\\ = \frac{{1.4}}{{2.3}}.\frac{{2.5}}{{3.4}}...\frac{{2013.2016}}{{2014.2015}}\\ = \frac{{1.2...2013}}{{2.3...2014}}.\frac{{4.5...2016}}{{3.4...2015}}\\ = \frac{1}{{2014}}.\frac{{2016}}{3} = \frac{{336}}{{1007}}\end{array}\]

\(\)\(\frac{{2014}}{{2016}}P = \frac{{2014}}{{2016}}.\frac{{337}}{{1007}} = \frac{1}{3}\)