\(\frac{1}{5\times9}+\frac{1}{9\times13}+\frac{1}{13\times17}+...+\frac{1}{41\times45}\)
= \(\frac{1}{4}\times\left(\frac{4}{5\times9}+\frac{4}{9\times13}+\frac{4}{13\times17}+...+\frac{4}{41\times45}\right)\)
= \(\frac{1}{4}\times\left(\frac{1}{5}-\frac{1}{9}+\frac{1}{9}-\frac{1}{13}+\frac{1}{13}-\frac{1}{17}+...+\frac{1}{41}-\frac{1}{45}\right)\)
= \(\frac{1}{4}\times\left(\frac{1}{5}-\frac{1}{45}\right)\)
= \(\frac{1}{4}\times\frac{8}{45}=\frac{2}{45}\)
\(A=\frac{1}{5\cdot9}+\frac{1}{9\cdot13}+...+\frac{1}{41\cdot45}\)
\(4A=\frac{4}{5\cdot9}+\frac{4}{9\cdot13}+...+\frac{4}{41\cdot45}\)
\(4A=\frac{1}{5}-\frac{1}{9}+\frac{1}{9}-\frac{1}{13}+...+\frac{1}{41}-\frac{1}{45}\)
\(4A=\frac{1}{5}-\frac{1}{45}\)
\(4A=\frac{8}{45}\)
\(A=\frac{2}{45}\)
