(0,5 điểm) Tính \(\frac{P}{Q},\) biết rằng:

\(P = \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + ...\frac{1}{{499}} + \frac{1}{{500}},\) \(Q = \frac{1}{{499}} + \frac{2}{{498}} + \frac{3}{{497}} + ... + \frac{{498}}{2} + \frac{{499}}{1}\).

Hướng dẫn giải

Xét \(Q = \frac{1}{{499}} + \frac{2}{{498}} + \frac{3}{{497}} + ... + \frac{{498}}{2} + \frac{{499}}{1}\)

\( = \left( {\frac{1}{{499}} + 1} \right) + \left( {\frac{2}{{498}} + 1} \right) + \left( {\frac{3}{{497}} + 1} \right) + ... + \left( {\frac{{498}}{2} + 1} \right) + 1\)

\( = \frac{{500}}{{499}} + \frac{{500}}{{498}} + \frac{{500}}{{497}} + ... + \frac{{500}}{2} + 1\)

\( = 500 \cdot \left( {\frac{1}{{499}} + \frac{1}{{498}} + \frac{1}{{497}} + ... + \frac{1}{2} + \frac{1}{{500}}} \right)\)

\( = 500 \cdot P\).

Do đó, \(\frac{P}{Q} = \frac{P}{{500 \cdot P}} = \frac{1}{{500}}.\)