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

Ta tính:

\(\frac{1}{{1 \times 99}} + \frac{1}{{3 \times 97}} + \frac{1}{{5 \times 95}} + \ldots + \frac{1}{{97 \times 3}} + \frac{1}{{99 \times 1}}\)

\( = \frac{2}{{1 \times 99}} + \frac{2}{{3 \times 97}} + \frac{2}{{5 \times 95}} \ldots + \frac{2}{{49 \times 51}}\)

\( = \frac{1}{{50}} \times (\frac{{50 \times 2}}{{1 \times 99}} + \frac{{50 \times 2}}{{3 \times 97}} + \frac{{50 \times 2}}{{5 \times 95}} + \ldots + \frac{{50 \times 2}}{{49 \times 51}})\)

\( = \frac{1}{{50}} \times (\frac{{100}}{{1 \times 99}} + \frac{{100}}{{3 \times 97}} + \frac{{100}}{{5 \times 95}} + \ldots + \frac{{100}}{{49 \times 51}})\)

\( = \frac{1}{{50}} \times (\frac{{1 + 99}}{{1 \times 99}} + \frac{{3 + 97}}{{3 \times 97}} + \frac{{5 + 95}}{{5 \times 95}} + \ldots + \frac{{49 + 51}}{{49 \times 51}})\)

\( = \frac{1}{{50}} \times \left( {\frac{1}{{1 \times 99}} + \frac{{99}}{{1 \times 99}} + \frac{3}{{3 \times 97}} + \frac{{97}}{{3 \times 97}} + \frac{5}{{5 \times 95}} + \frac{{95}}{{5 \times 95}} + \ldots + \frac{{49}}{{49 \times 51}} + \frac{{51}}{{49 \times 51}}} \right)\)

\( = \frac{1}{{50}} \times \left( {\frac{1}{{99}} + 1 + \frac{1}{{97}} + \frac{1}{3} + \frac{1}{{95}} + \frac{1}{5} + \ldots + \frac{1}{{49}} + \frac{1}{{51}}} \right)\)

\( = \frac{1}{{50}} \times (1 + \frac{1}{3} + \frac{1}{5} + \ldots + \frac{1}{{97}} + \frac{1}{{99}})\)

Vậy:

\(\frac{{1 + \frac{1}{3} + \frac{1}{5} + \ldots + \frac{1}{{97}} + \frac{1}{{99}}}}{{\frac{1}{{1 \times 99}} + \frac{1}{{3 \times 97}} + \frac{1}{{5 \times 95}} + \ldots + \frac{1}{{97 \times 3}} + \frac{1}{{99 \times 1}}}} = \frac{{1 + \frac{1}{3} + \frac{1}{5} + \ldots + \frac{1}{{97}} + \frac{1}{{99}}}}{{\frac{1}{{50}} \times (1 + \frac{1}{3} + \frac{1}{5} + \ldots + \frac{1}{{97}} + \frac{1}{{99}})}} = \frac{1}{{\frac{1}{{50}}}} = 50\)Đáp Số: 50.