\(\frac{1}{5}+\frac{1}{45}+\frac{1}{117}+\frac{1}{221}+...+\frac{1}{9021}+\frac{1}{9797}\)
\(=\frac{1}{1.5}+\frac{1}{5.9}+\frac{1}{9.13}+\frac{1}{13.17}+...+\frac{1}{93.97}+\frac{1}{97.101}\)
\(=\frac{1}{4}.\left(\frac{4}{1.5}+\frac{4}{5.9}+\frac{4}{9.13}+\frac{4}{13.17}+...+\frac{4}{93.97}+\frac{4}{97.101}\right)\)
\(=\frac{1}{4}.\left(1-\frac{1}{5}+\frac{1}{5}-\frac{1}{9}+...+\frac{1}{97}-\frac{1}{101}\right)\)
\(=\frac{1}{4}.\left(1-\frac{1}{101}\right)\)
\(=\frac{1}{4}.\left(\frac{101}{101}-\frac{1}{101}\right)\)
\(=\frac{1}{4}.\frac{100}{101}\)
\(=\frac{25}{101}\)
1/5+1/45+1/117+1/221+...+1/9021+1/9797
=1/1.5+1/5.9+1/9.13+1/13.17+...+1/93.97+1/97.101
=1-1/5+1/5-1/9+1/9-1/13+1/13-1/17+...+1/93-1/97+1/97-1/101
=1-1/101
=100/101
