Tính nhanh:

\(A = \frac{{2017 + \frac{{2016}}{2} + \frac{{2015}}{3} + .... + \frac{1}{{2017}} + 2017}}{{1 + \frac{1}{2} + \frac{1}{3} + ... + \frac{1}{{2017}}}}\)

Hướng Dẫn Giải

Đặt \(A = 1 + \frac{1}{3} + \frac{1}{9} + \frac{1}{{27}} + \frac{1}{{81}} + \frac{1}{{243}} + \frac{1}{{729}}\)

Suy ra: \(3 \times A = 3 \times (1 + \frac{1}{3} + \frac{1}{9} + \frac{1}{{27}} + \frac{1}{{81}} + \frac{1}{{243}} + \frac{1}{{729}})\)

\( = 3 + 1 + \frac{1}{3} + \frac{1}{9} + \frac{1}{{27}} + \frac{1}{{81}} + \frac{1}{{243}}\).

Do đó:

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

\( \to 2 \times A = 3 - \frac{1}{{729}} = \frac{{2187}}{{729}} - \frac{1}{{729}} = \frac{{2186}}{{729}}\)

\( \to A = \frac{{2186}}{{729}}:2 = \frac{{1093}}{{729}}\)

Đáp Số: \(A = \frac{{1093}}{{729}}\).

Hướng Dẫn Giải

\((1 - \frac{3}{4}) \times (1 - \frac{3}{7}) \times (1 - \frac{3}{{10}}) \times \ldots \times (1 - \frac{3}{{100}}) = \ldots \)

\( = \frac{1}{4} \times \frac{4}{7} \times \frac{7}{{10}} \times \ldots \times \frac{{97}}{{100}} = \frac{1}{{100}}\)

Đáp Số: \(\frac{1}{{100}}\)