Đáp án:
$A=\dfrac{200}{101}\\
B=\dfrac{50}{101}$
Giải thích các bước giải:
$A=\left ( 1+\dfrac{1}{1.3} \right ).\left ( 1+\dfrac{1}{2.4} \right ).\left ( 1+\dfrac{1}{3.5} \right )....\left ( 1+\dfrac{1}{99.101} \right )\\
=\dfrac{4}{1.3}.\dfrac{9}{2.4}.\dfrac{16}{3.5}....\dfrac{10000}{99.101}\\
=\dfrac{2.2}{1.3}.\dfrac{3.3}{2.4}.\dfrac{4.4}{3.5}....\dfrac{100.100}{99.101}\\
=\dfrac{2.3.4....100}{1.2.3...99}.\dfrac{2.3.4...100}{3.4.5...101}\\
=\dfrac{100}{1}.\dfrac{2}{101}\\
=\dfrac{200}{101}\\
B=\dfrac{1}{1.3}.\dfrac{9}{2.4}.\dfrac{16}{3.5}...\dfrac{10000}{99.101}\\
=\dfrac{1}{1.3}.\dfrac{3.3}{2.4}.\dfrac{4.4}{3.5}...\dfrac{100.100}{99.101}\\
=\dfrac{1.3.4...100}{1.2.3...99}.\dfrac{1.3.4...100}{3.4.5...101}\\
=\dfrac{100}{2}.\dfrac{1}{101}\\
=\dfrac{50}{101}$
