\(\Rightarrow A=\frac{1}{3\times5}+\frac{1}{5\times7}+.......+\frac{1}{99\times101}\)
\(\Rightarrow A=\frac{1}{2}\times\left(\frac{2}{3\times5}+\frac{2}{5\times7}+........+\frac{2}{99\times101}\right)\)
\(\Rightarrow A=\frac{1}{2}\times\left(\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}+........+\frac{1}{99}-\frac{1}{101}\right)\)
\(\Rightarrow A=\frac{1}{2}\times\left(\frac{1}{3}-\frac{1}{101}\right)\)
\(\Rightarrow A=\frac{1}{2}\times\frac{98}{303}\)
\(\Rightarrow A=\frac{49}{303}\)
A = \(\frac{1}{15}+\frac{1}{35}+...+\frac{1}{9999}\)
= \(\frac{1}{3x5}+\frac{1}{5x7}+...+\frac{1}{99x101}\)
= \(\frac{1}{2}x\left(\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}+...+\frac{1}{99}-\frac{1}{101}\right)\)
= \(\frac{1}{2}x\left(\frac{1}{3}-\frac{1}{101}\right)\)
= \(\frac{1}{2}x\frac{98}{303}\)
= \(\frac{49}{303}\)
A = 1/3x5 + 1/5x7 + 1/7x9 + 1/9x11 + .....+ 1/99x101
A = ( 1/3x5 + 1/5x7 + 1/7x9 + 1/9x11 + .....+ 1/99x101) x 2 x 1/2
A = ( 2/3x5 + 2/5x7 + 2/7x9 + 2/9x11 + .....+ 2/99x101) x 1/2
A = ( 1/3 -1/5 + 1/5 - 1/7 + 1/7 - 1/9 + 1/9 - 1/11 + ..... + 1/99 - 1/101) x 1/2
A = ( 1/3 - 1/101 ) x 1/2
A = 98/303 x 1/2
A = 49/303
