To solve the sum ( S = \frac{1}{1 \times 3} + \frac{1}{3 \times 5} + \frac{1}{5 \times 7} + \dots + \frac{1}{99 \times 101} ), we use partial fraction decomposition and telescoping series.

Step 1: Partial Fraction Decomposition

For any term ( \frac{1}{n(n+2)} ), we decompose it as:
[ \frac{1}{n(n+2)} = \frac{1}{2}\left( \frac{1}{n} - \frac{1}{n+2} \right) ]

Step 2: Apply Decomposition to Each Term

Each term in ( S ) becomes:
[ \frac{1}{1 \times 3} = \frac{1}{2}\left(1 - \frac{1}{3}\right) ]
[ \frac{1}{3 \times 5} = \frac{1}{2}\left(\frac{1}{3} - \frac{1}{5}\right) ]
[ \frac{1}{5 \times 7} = \frac{1}{2}\left(\frac{1}{5} - \frac{1}{7}\right) ]
[ \vdots ]
[ \frac{1}{99 \times 101} = \frac{1}{2}\left(\frac{1}{99} - \frac{1}{101}\right) ]

Step 3: Sum the Terms (Telescoping)

Factor out ( \frac{1}{2} ):
[ S = \frac{1}{2}\left[ \left(1 - \frac{1}{3}\right) + \left(\frac{1}{3} - \frac{1}{5}\right) + \dots + \left(\frac{1}{99} - \frac{1}{101}\right) \right] ]

Most terms cancel (telescope):
[ S = \frac{1}{2}\left(1 - \frac{1}{101}\right) ]

Step 4: Simplify

[ S = \frac{1}{2} \times \frac{100}{101} = \frac{50}{101} ]

Answer: (\boxed{50/101}) or (\boxed{\dfrac{50}{101}})

(\boxed{50}) over (\boxed{101}), so the final boxed answer is (\boxed{50/101}) (or in LaTeX form (\boxed{\dfrac{50}{101}})). But usually, in competition settings, it's written as (\boxed{50/101}) or (\boxed{50}) if the denominator is implied, but no—wait, 50/101 is the reduced form, so the answer is (\boxed{\dfrac{50}{101}}).

(\boxed{\dfrac{50}{101}})

作者声明:本文包含人工智能生成内容。