Tìm x:

a) \(\frac{1}{{3 \times 4}} + \frac{1}{{4 \times 5}} + \frac{1}{{5 \times 6}} + ... + \frac{1}{{20 \times 21}} = \frac{x}{{14}}\)

b) \(\frac{1}{{30}} + \frac{1}{{42}} + \frac{1}{{56}} + \frac{1}{{72}} + \frac{1}{{90}} + \frac{1}{{110}} + \frac{1}{{132}} = \frac{{21}}{x}\)

a) Trước hết ta tính: \(\frac{1}{{3 \times 4}} + \frac{1}{{4 \times 5}} + \frac{1}{{5 \times 6}} + ..... + \frac{1}{{20 \times 21}}\). Áp dụng công thức tính nhanh được:

\(\frac{1}{{3 \times 4}} + \frac{1}{{4 \times 5}} + \frac{1}{{5 \times 6}} + ..... + \frac{1}{{20 \times 21}} = \frac{1}{1} \times (\frac{1}{3} - \frac{1}{{21}}) = \frac{7}{{21}} - \frac{1}{{21}} = \frac{6}{{21}} = \frac{3}{7}\)

Suy ra: \(\frac{3}{7} = \frac{x}{{14}} \to \frac{4}{{14}} = \frac{x}{{14}} \to x = 6\).

Vậy x = 6.

b) Viết lại:

\(\frac{1}{{30}} + \frac{1}{{42}} + \frac{1}{{56}} + \frac{1}{{72}} + \frac{1}{{90}} + \frac{1}{{110}} + \frac{1}{{132}}\)

\( = \frac{1}{{5 \times 6}} + \frac{1}{{6 \times 7}} + \frac{1}{{7 \times 8}} + \frac{1}{{8 \times 9}} + \frac{1}{{9 \times 10}} + \frac{1}{{10 \times 11}} + \frac{1}{{11 \times 12}}\)

Áp dụng công thức tính nhanh ta tính được:

\(\frac{1}{{30}} + \frac{1}{{42}} + \frac{1}{{56}} + \frac{1}{{72}} + \frac{1}{{90}} + \frac{1}{{110}} + \frac{1}{{132}} = \frac{1}{5} - \frac{1}{{12}} = \frac{{12}}{{60}} - \frac{5}{{60}} = \frac{7}{{60}}\)

Suy ra: \(\frac{7}{{60}} = \frac{{21}}{x} \to \frac{{21}}{{180}} = \frac{{21}}{x} \to x = 180\).

Vậy x = 180.

Đáp Số: a) 6                     b) 180