Tính giá trị biểu thức: P = 1/(1.3) + 1/(3.5) + ... + 1/(99.101)

\(P=\frac{1}{1.3}+\frac{1}{3.5}+...+\ \frac{1}{99.101}\)

\(P=\frac{1}{2}.\ \left(\frac{2}{1.3}+\frac{2}{3.5}+...+\frac{2}{99.101}\right)\)

\(P=\frac{1}{2}.\ \left(1-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+...+\frac{1}{99}-\frac{1}{101}\right)\)

\(P=\frac{1}{2}.\ \left(1-\frac{1}{101}\right)\)

\(P=\frac{1}{2}.\ \frac{100}{101}=\frac{50}{101}\)

\(P=\frac{1}{1.3}+\frac{1}{3.5}+\frac{1}{5.7\ }+...+\frac{1}{99.101}\)

\(2P=2.\left(\frac{1}{1.3}+\frac{1}{3.5}+\frac{1}{5.7\ }+...+\frac{1}{99.101}\right)\)

\(2P=\frac{2}{1.3}+\frac{2}{3.5}+\frac{2}{5.7\ }+...+\frac{2}{99.101}\)

\(2P=1-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}+...+\frac{1}{99}-\frac{1}{101}\)

\(2P=1-\frac{1}{101}\)

\(2P=\frac{100}{101}\)

\(P=\frac{100}{101}:2=\frac{50}{101}\)

P = 1/(1.3) + 1/(3.5) + ... + 1/(99.101)

= 1/2 . [2/(1.3) + 2/(3.5) + ... + 2/(99.101)]

= 1/2 . (1 - 1/3 + 1/3 - 1/5 +....+ 1/99 - 1/101)

= 1/2 . ( 1- 1/101)

= 1/2 . 100/101

= 50/101