Tính (1-1/1 2)x(1-1/1 2 3)x(1-1/1 2 3 4)x....x(1-1/1 2 3 ... 2016)

Lời giải:

Ta có: 1+ 2  + 3 +…+ n = n(n + 1) : 2

\(\left( {1 - \frac{1}{{1 + 2}}} \right)\left( {1 - \frac{1}{{1 + 2 + 3}}} \right)...\left( {1 - \frac{1}{{1 + 2 + ... + 2016}}} \right)\)

\( = \left( {1 - \frac{1}{{(1 + 2)2:2}}} \right)\left( {1 - \frac{1}{{(1 + 3)3:2}}} \right)\left( {1 - \frac{1}{{(1 + 4)4:2}}} \right)...\left( {1 - \frac{1}{{(1 + 2016)2016:2}}} \right)\)

\( = \left( {1 - \frac{2}{{2.3}}} \right)\left( {1 - \frac{2}{{3.4}}} \right)...\left( {1 - \frac{2}{{2016.2017}}} \right)\)

\( = \frac{{2.3 - 2}}{{2.3}}.\frac{{3.4 - 2}}{{3.4}}...\frac{{2016.2017 - 2}}{{2016.2017}}\)

\( = \frac{{1.4}}{{2.3}}.\frac{{2.5}}{{3.4}}...\frac{{2015.2018}}{{2016.2017}}\)

\( = \frac{1}{3}.\frac{{2018}}{{2016}} = \frac{{2018}}{{6048}}\).