Tính nhanh giá trị của biểu thức: B = 4/( 1 ⋅ 3 ⋅ 5) + 4/( 3 ⋅ 5 ⋅ 7) + 4/( 5 ⋅ 7 ⋅ 9) + . . . + 4/( 21 ⋅ 23 ⋅ 25) .

Hướng dẫn giải:

\(B = \frac{4}{{1 \cdot 3 \cdot 5}} + \frac{4}{{3 \cdot 5 \cdot 7}} + \frac{4}{{5 \cdot 7 \cdot 9}} + ... + \frac{4}{{21 \cdot 23 \cdot 25}}\)

\( = \frac{{5 - 1}}{{1 \cdot 3 \cdot 5}} + \frac{{7 - 3}}{{3 \cdot 5 \cdot 7}} + \frac{{9 - 5}}{{5 \cdot 7 \cdot 9}} + ... + \frac{{25 - 21}}{{21 \cdot 23 \cdot 25}}\)

\[ = \frac{5}{{1 \cdot 3 \cdot 5}} - \frac{1}{{1 \cdot 3 \cdot 5}} + \frac{7}{{3 \cdot 5 \cdot 7}} - \frac{3}{{3 \cdot 5 \cdot 7}} + \frac{9}{{5 \cdot 7 \cdot 9}} - \frac{5}{{5 \cdot 7 \cdot 9}} + ... + \frac{{25}}{{21 \cdot 23 \cdot 25}} - \frac{{21}}{{21 \cdot 23 \cdot 25}}\]

\( = \frac{1}{{1 \cdot 3}} - \frac{1}{{3 \cdot 5}} + \frac{1}{{3 \cdot 5}} - \frac{1}{{5 \cdot 7}} + \frac{1}{{5 \cdot 7}} - \frac{1}{{7 \cdot 9}} + ... + \frac{1}{{21 \cdot 22}} - \frac{1}{{22 \cdot 23}}\)

\[ = \frac{1}{{1 \cdot 3}} - \frac{1}{{22 \cdot 23}} = \frac{1}{3} - \frac{1}{{506}} = \frac{{506}}{{1\,\,518}} - \frac{3}{{1\,\,518}} = \frac{{503}}{{1\,\,518}}.\]