Đáp án:
\(\begin{array}{l}
A = \frac{{99}}{{101}}\\
B = - \frac{1}{{64}}
\end{array}\)
Giải thích các bước giải:
\(\begin{array}{l}
a)\,\,\,A = \frac{1}{{1 + 2}} + \frac{1}{{1 + 2 + 3}} + ..... + \frac{1}{{1 + 2 + 3 + ... + 100}}\\
Ap\,\,dung\,\,cong\,\,thuc:\\
1 + 2 + 3 + ... + n = \frac{{n\left( {n + 1} \right)}}{2}\\
\Rightarrow A = \frac{1}{{\frac{{2.3}}{2}}} + \frac{1}{{\frac{{3.4}}{2}}} + \frac{1}{{\frac{{4.5}}{2}}} + .... + \frac{1}{{\frac{{100.101}}{2}}}\\
= \frac{2}{{2.3}} + \frac{2}{{3.4}} + \frac{2}{{4.5}} + .... + \frac{2}{{100.101}}\\
= 2\left( {\frac{1}{{2.3}} + \frac{1}{{3.4}} + .... + \frac{1}{{100.101}}} \right)\\
= 2\left( {\frac{1}{2} - \frac{1}{3} + \frac{1}{3} - \frac{1}{4} + ... + \frac{1}{{100}} - \frac{1}{{101}}} \right)\\
= 2\left( {\frac{1}{2} - \frac{1}{{101}}} \right) = \frac{{2\left( {101 - 2} \right)}}{{2.101}} = \frac{{99}}{{101}}.\\
b)\,\,\,B = \frac{3}{2} + \frac{5}{4} + \frac{9}{8} + \frac{{17}}{{16}} + \frac{{33}}{{32}} + \frac{{65}}{{64}} - 7\\
= \frac{3}{2} - 1 + \frac{5}{4} - 1 + \frac{9}{8} - 1 + \frac{{17}}{{16}} - 1 + \frac{{33}}{{32}} - 1 + \frac{{65}}{{64}} - 1 - 1\\
= \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{{16}} + \frac{1}{{32}} + \frac{1}{{64}} - 1\\
= \frac{3}{4} + \frac{3}{{16}} + \frac{3}{{64}} - 1\\
= 3.\frac{{16 + 4 + 1}}{{64}} - 1\\
= \frac{{3.21 - 64}}{{64}} = - \frac{1}{{64}}.
\end{array}\)
