Tính nhanh:

a) \(A = \frac{{64 \times 50 + 100 \times 44}}{{27 \times 38 + 146 \times 19}}\)

b) \(B = \frac{{155 + 818 + 45 + 182}}{{999 - 77 + 301 - 23}}\)

c) \(C = \frac{{2007 \times 2006 - 1}}{{2005 \times 2007 + 2006}}\)

d) \(D = \frac{1}{3} + \frac{1}{9} + \frac{1}{{27}} + \frac{1}{{81}} + \frac{1}{{243}} + \frac{1}{{729}}\)

e) \(E = \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{{16}} + \frac{1}{{32}} + \frac{1}{{64}}\)

f) \(F = \frac{1}{{2 \times 4}} + \frac{1}{{4 \times 6}} + \frac{1}{{6 \times 8}} + ... + \frac{1}{{98 \times 100}}\)

g) \(G = \frac{5}{6} + \frac{{11}}{{12}} + \frac{{19}}{{20}} + \frac{{29}}{{30}} + \frac{{41}}{{42}} + \frac{{55}}{{56}} + \frac{{71}}{{72}} + \frac{{89}}{{90}}\)

a) \(A = \frac{{64 \times 50 + 100 \times 44}}{{27 \times 38 + 146 \times 19}} = \frac{{50 \times \left( {64 + 44 \times 2} \right)}}{{19 \times \left( {27 \times 2 + 146} \right)}} = \frac{{50 \times 152}}{{19 \times 200}} = \frac{{20 \times 19 \times 8}}{{19 \times 50 \times 4}} = 2\)

b) \(B = \frac{{155 + 818 + 45 + 182}}{{999 - 77 + 301 - 23}} = \frac{{\left( {155 + 45} \right) + \left( {818 + 182} \right)}}{{\left( {999 + 301} \right) - \left( {77 + 23} \right)}} = \frac{{200 + 1000}}{{1300 - 100}} = \frac{{1200}}{{1200}} = 1\)

c) \(C = \frac{{2007 \times 2006 - 1}}{{2005 \times 2007 + 2006}} = \frac{{2007 \times \left( {2005 + 1} \right) - 1}}{{2006 + 2005 \times 2007}} = \frac{{2005 \times 2007 + 2007 - 1}}{{2006 + 2005 \times 2007}}\)

\( = \frac{{2005 \times 2007 + 2006}}{{2005 \times 2007 + 2006}} = 1\)

d) \(D = \frac{1}{3} + \frac{1}{9} + \frac{1}{{27}} + \frac{1}{{81}} + \frac{1}{{243}} + \frac{1}{{729}}\)

\(3 \times D = 3 \times \left( {\frac{1}{3} + \frac{1}{9} + \frac{1}{{27}} + \frac{1}{{81}} + \frac{1}{{243}} + \frac{1}{{729}}} \right)\)\( = 1 + \frac{1}{3} + \frac{1}{9} + \frac{1}{{27}} + \frac{1}{{81}} + \frac{1}{{243}}\)

\(3 \times D - D = 1 + \frac{1}{3} + \frac{1}{9} + \frac{1}{{27}} + \frac{1}{{81}} + \frac{1}{{243}} - \left( {\frac{1}{3} + \frac{1}{9} + \frac{1}{{27}} + \frac{1}{{81}} + \frac{1}{{243}} + \frac{1}{{729}}} \right)\)

\(2 \times D = 1 - \frac{1}{{729}} = \frac{{728}}{{729}}\)

\(D = \frac{{728}}{{729}}:2 = \frac{{364}}{{729}}\)

e) \(E = \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{{16}} + \frac{1}{{32}} + \frac{1}{{64}}\)

\(2 \times E = 2 \times \left( {\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{{16}} + \frac{1}{{32}} + \frac{1}{{64}}} \right)\)\( = 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{{16}} + \frac{1}{{32}}\)

\(2 \times E - E = 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{{16}} + \frac{1}{{32}} - \left( {\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{{16}} + \frac{1}{{32}} + \frac{1}{{64}}} \right)\)

\(E = 1 - \frac{1}{{64}} = \frac{{63}}{{64}}\)

f) \(F = \frac{1}{{2 \times 4}} + \frac{1}{{4 \times 6}} + \frac{1}{{6 \times 8}} + ... + \frac{1}{{98 \times 100}}\)

\(F = \frac{1}{2} \times \left( {\frac{2}{{2 \times 4}} + \frac{2}{{4 \times 6}} + \frac{2}{{6 \times 8}} + ... + \frac{2}{{98 \times 100}}} \right)\)

\(F = \frac{1}{2} \times \left( {\frac{1}{2} - \frac{1}{4} + \frac{1}{4} - \frac{1}{6} + \frac{1}{6} - \frac{1}{8} + ... + \frac{1}{{98}} - \frac{1}{{100}}} \right)\)

\(F = \frac{1}{2} \times \left( {\frac{1}{2} - \frac{1}{{100}}} \right) = \frac{1}{2} \times \frac{{49}}{{100}} = \frac{{49}}{{200}}\)

g) \(G = \frac{5}{6} + \frac{{11}}{{12}} + \frac{{19}}{{20}} + \frac{{29}}{{30}} + \frac{{41}}{{42}} + \frac{{55}}{{56}} + \frac{{71}}{{72}} + \frac{{89}}{{90}}\)

\(G = 1 - \frac{1}{6} + 1 - \frac{1}{{12}} + 1 - \frac{1}{{20}} + 1 - \frac{1}{{30}} + 1 - \frac{1}{{42}} + 1 - \frac{1}{{56}} + 1 - \frac{1}{{72}} + 1 - \frac{1}{{90}}\)

\(G = 8 - \left( {\frac{1}{6} + \frac{1}{{12}} + \frac{1}{{20}} + \frac{1}{{30}} + \frac{1}{{42}} + \frac{1}{{56}} + \frac{1}{{72}} + \frac{1}{{90}}} \right)\)

\(G = 8 - \left( {\frac{1}{{2 \times 3}} + \frac{1}{{3 \times 4}} + \frac{1}{{4 \times 5}} + \frac{1}{{5 \times 6}} + \frac{1}{{6 \times 7}} + \frac{1}{{7 \times 8}} + \frac{1}{{8 \times 9}} + \frac{1}{{9 \times 10}}} \right)\)

\(G = 8 - \left( {\frac{1}{2} - \frac{1}{{10}}} \right)\)

\(G = 8 - \frac{2}{5} = \frac{{38}}{5}\)