Tính:

a) \(\frac{1}{{2 \times 3}} + \frac{1}{{3 \times 4}} + \frac{1}{{4 \times 5}} + ... + \frac{1}{{19 \times 20}} = ......\)

b) \(\frac{1}{2} + \frac{1}{6} + \frac{1}{{12}} + ... + \frac{1}{{72}} + \frac{1}{{90}} = ......\)

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: \(\frac{1}{{100}}\)