Câu 12. Cho $A = \frac{1}{{1 \times 300}} + \frac{1}{{2 \times 301}} + \frac{1}{{3 \times 302}} + .... + \frac{1}{{101 \times 400}}$ và Tính tỉ số $\frac{A}{B}$ $B = \frac{1}{{1 \times 102}} + \frac{1}{{2 \times 103}} + \frac{1}{{3 \times 104}} + .... + \frac{1}{{299 \times 400}}$

Ta có $A = \frac{1}{{1 \times 300}} + \frac{1}{{2 \times 301}} + \frac{1}{{3 \times 302}} + .... + \frac{1}{{101 \times 400}}$

$299 \times A = \frac{{299}}{{1 \times 300}} + \frac{{299}}{{2 \times 301}} + \frac{{299}}{{3 \times 302}} + .... + \frac{{299}}{{101 \times 400}}$

$299 \times A = 1 - \frac{1}{{300}} + \frac{1}{2} - \frac{1}{{301}} + \frac{1}{3} - \frac{1}{{302}} + .... + \frac{1}{{101}} - \frac{1}{{400}}$

$299 \times A = (1 + \frac{1}{2} + \frac{1}{3} + ... + \frac{1}{{101}}) - (\frac{1}{{300}} + \frac{1}{{301}} + \frac{1}{{302}} + ... + \frac{1}{{400}})$

Ta có $B = \frac{1}{{1 \times 102}} + \frac{1}{{2 \times 103}} + \frac{1}{{3 \times 104}} + .... + \frac{1}{{299 \times 400}}$

$101 \times B = \frac{{101}}{{1 \times 102}} + \frac{{101}}{{2 \times 103}} + \frac{{101}}{{3 \times 104}} + ... + \frac{{101}}{{299 \times 400}}$

$101 \times B = 1 - \frac{1}{{102}} + \frac{1}{2} - \frac{1}{{103}} + \frac{1}{3} - \frac{1}{{104}} + ... + \frac{1}{{299}} - \frac{1}{{400}}$

$101 \times B = \left( {1 + \frac{1}{2} + \frac{1}{3} + ...\frac{1}{{299}}} \right) - \left( {\frac{1}{{102}} + \frac{1}{{103}} + \frac{1}{{104}} + ... + \frac{1}{{400}}} \right)$

$101 \times B = \left( {1 + \frac{1}{2} + \frac{1}{3} + ... + \frac{1}{{101}}} \right) - \left( {\frac{1}{{300}} + \frac{1}{{301}} + \frac{1}{{302}} + ... + \frac{1}{{400}}} \right)$

Suy ra 299 x A = 101 x B hay $\frac{A}{B} = \frac{{101}}{{299}}$