Đáp án:
Giải thích các bước giải:
\(A = 124 \cdot \left( {\frac{1}{{1.1985}} + \frac{1}{{2.1986}} + \frac{1}{{3.1987}} + ... + \frac{1}{{16.2000}}} \right)\)
\( = \frac{{124}}{{1984}} \cdot \left( {\frac{{1984}}{{1.1985}} + \frac{{1984}}{{2.1986}} + \frac{{1984}}{{3.1987}} + ... + \frac{{1984}}{{16.2000}}} \right)\)
\(= \frac{1}{{16}}\left( {1 - \frac{1}{{1985}} + \frac{1}{2} - \frac{1}{{1986}} + \frac{1}{3} - \frac{1}{{1987}} + ... + \frac{1}{{16}} - \frac{1}{{2000}}} \right)\)
\( = \frac{1}{{16}} \cdot \left( {1 + \frac{1}{2} + \frac{1}{3} + ... + \frac{1}{{16}}} \right) \cdot \left( {\frac{1}{{1985}} + \frac{1}{{1986}} + \frac{1}{{1987}} + ... + \frac{1}{{2000}}} \right)\)
\(B = \frac{1}{{1.17}} + \frac{1}{{2.18}} + ... + \frac{1}{{1984.2000}}\)
\(= \frac{1}{{16}}\left( {\frac{{16}}{{1.17}} + \frac{{16}}{{2.18}} + ... + \frac{{16}}{{1984.2000}}} \right)\)
\(= \frac{1}{{16}}\left( {1 - \frac{1}{{17}} + \frac{1}{2} - \frac{1}{{18}} + ... + \frac{1}{{1984}} - \frac{1}{{2000}}} \right)\)
\( = \frac{1}{{16}}\left[ {\left( {1 + \frac{1}{2} + ... + \frac{1}{{1984}}} \right) - \left( {\frac{1}{{17}} + \frac{1}{{18}} + ... + \frac{1}{{2000}}} \right)} \right]\)
\( = \frac{1}{{16}}\left( {1 + \frac{1}{2} + ... + \frac{1}{{16}}} \right) + \left( {\frac{1}{{17}} + \frac{1}{{18}} + ... + \frac{1}{{1984}}} \right) - \left( {\frac{1}{{17}} + \frac{1}{{18}} + ... + \frac{1}{{1984}}} \right) - \left( {\frac{1}{{1985}} + \frac{1}{{1986}} + ... + \frac{1}{{2000}}} \right)\)
\(= \frac{1}{{16}}\left[ {\left( {1 + \frac{1}{2} + ... + \frac{1}{{16}}} \right) - \left( {\frac{1}{{1985}} + \frac{1}{{1986}} + ... + \frac{1}{{2000}}} \right)} \right]\)
Vậy \(A=B\).
